Apps/MaclaurinToroid

Maclaurin Toroid (MPT77)

Maclaurin Toroid Sequence 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,

$\dot{\phi }{\varpi }^2$	$=$	$\frac{5L}{2M}\{1-[1-m(\varpi ){]}^{2/3}\},$
📚 Stoeckly (1965), §II.c, Eq. (12) 📚 Ostriker & Mark (1968), Eq. (45) 📚 Bodenheimer & Ostriker (1970), Eq. (12) 📚 Bodenheimer & Ostriker (1973), Eq. (3) 📚 Eriguchi & Hachisu (1985), Eq. (1)

where, $L$ is the total angular momentum, $M$ is the total mass, 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, although MPT77 do not use this terminology. 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 shows how $L_{*}$ varies with $\tau$ along the Maclaurin Spheroid sequence. In an effort to conform to MPT77's presentation, our Figure 1 (immediately below) displays the same information as displayed in Figure 4 of this separate chapter, but the axes have been swapped and the maximum displayed value of $L_{*}$ has been extended from 1 to 1.5.

EFE Diagram   Figure 1   Figure 2 This multicolor curve also appears as a solid black curve in: Fig. 5 (p. 594) of 📚 Marcus, Press, & Teukolsky (1977) This multicolor curve also appears as a solid black curve in: Fig. 4 (p. 593) of 📚 Marcus, Press, & Teukolsky (1977)

Constructed Maclaurin Toroid Models

📚 Marcus, Press, & Teukolsky (1977) did not create a tabulated description of the models that they constructed along their so-called "Maclaurin Toroid" sequence. Throughout their paper, however, they identify the properties of a selected group of equilibrium models. Here is a list of the Maclaurin Toroid models that we have pulled from their discussion.

Model	$L_{*}$	Spheroid Equivalent			Notes …
		$e$	$\tau$	$E_{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{5}{3}(T+W)/E_{\mathrm{T}\mathrm{7}\mathrm{8}}$
$L^{-}$	$\sim 0.775$	$\sim 0.99409$	$\sim 0.40585$	$-0.41685$	Toroid does not exist
$L_0$	$0.792\pm 0.002$	$0.9949\pm 0.0001$	$0.41195$	$-0.40439$	Total energy of toroid is same as the total energy of Maclaurin spheroid with same $L_{*}$.
	$0.8732$	$0.9975$	$0.4367$	$-0.35023$	Marginally stable Maclaurin spheroid and associated toroid; see MPT77's Figure 2 (p. 592). Also, one (of five) meridional cross-sections displayed in MPT77's Figure 3 (p. 593)
$L^{+}$	$0.965175$	$0.99892$	$0.45747$	$-0.29762$	Analytically known (!) onset of dynamical instability along Maclaurin spheroid sequence; see § 33 of EFE and the last row of Table B.1 from 📚 Bardeen (1971)
	$0.9852$	$0.99910$	$0.46104$	$-0.28754$	Four (of five) meridional cross-sections displayed in MPT77's Figure 3 (p. 593)
	$1.0731$	$0.99960$	$0.47369$	$-0.24745$
	$1.1489$	$0.99980$	$0.48125$	$-0.21841$
	$1.2262$	$0.9998999$	$0.486665$	$-0.19329$

Figure 3   Figure 4 Compare to: Fig. 5 (p. 594) of 📚 Marcus, Press, & Teukolsky (1977) Compare to: Fig. 4 (p. 593) of 📚 Marcus, Press, & Teukolsky (1977)

Maclaurin Toroid (EH85)

📚 Y. Eriguchi & I. Hachisu (1985, A&A, Vol. 148, pp. 289 - 292) have constructed a set of uniform-density, axisymmetric configurations that show how the Maclaurin toroid sequence is connected to the Maclaurin spheroid sequence. The following table displays the structural characteristics of these configurations; these numbers have been drawn directly from Table 1 of 📚 Eriguchi & Hachisu (1985).

Data extracted from 📚 Eriguchi & Hachisu (1985)				Our Determination
$h_0^2$	$j^2$	$\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}}{|W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}|}$	$\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_0}$	$L_{*}\equiv (4\pi /3{)}^{2/3}(3j^2{)}^{1/2}$
$5.802\times 10^{-2}$	$3.964\times 10^{-2}$	$0.445$	$-1.03\times 10^{-3}$	$0.8961$
$5.834\times 10^{-2}$	$3.816\times 10^{-2}$	$0.439$	$-1.02\times 10^{-3}$	$0.8792$
$5.916\times 10^{-2}$	$3.752\times 10^{-2}$	$0.437$	$-1.02\times 10^{-3}$	$0.8718$
$6.075\times 10^{-2}$	$3.718\times 10^{-2}$	$0.436$	$-1.02\times 10^{-3}$	$0.8678$
$6.416\times 10^{-2}$	$3.209\times 10^{-2}$	$0.412$	$-9.73\times 10^{-4}$	$0.8063$
$6.766\times 10^{-2}$	$3.090\times 10^{-2}$	$0.403$	$-9.61\times 10^{-4}$	$0.7912$
$7.070\times 10^{-2}$	$3.016\times 10^{-2}$	$0.389$	$-9.53\times 10^{-4}$	$0.7816$
$6.739\times 10^{-2}$	$3.192\times 10^{-2}$	$0.376$	$-9.74\times 10^{-4}$	$0.8041$
$5.376\times 10^{-2}$	$3.751\times 10^{-2}$	$0.368$	$-1.04\times 10^{-3}$	$0.8717$
$4.007\times 10^{-2}$	$4.502\times 10^{-2}$	$0.365$	$-1.12\times 10^{-3}$	$0.9550$
$3.202\times 10^{-2}$	$5.100\times 10^{-2}$	$0.366$	$-1.18\times 10^{-3}$	$1.0164$
$2.525\times 10^{-2}$	$5.975\times 10^{-2}$	$0.369$	$-1.26\times 10^{-3}$	$1.1002$
$1.811\times 10^{-2}$	$7.364\times 10^{-2}$	$0.378$	$-1.37\times 10^{-3}$	$1.2214$
$1.127\times 10^{-2}$	$9.914\times 10^{-2}$	$0.396$	$-1.53\times 10^{-3}$	$1.4171$
$4.812\times 10^{-3}$	$1.618\times 10^{-1}$	$0.430$	$-1.76\times 10^{-3}$	$1.8104$

See Also