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)

MPT77 also evaluate the normalized total energy, $E_{\mathrm{t}\mathrm{o}\mathrm{t}}/|E_0|$, of each of their constructed equilibrium configurations, where

$E_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$\equiv$

$T_{\mathrm{r}\mathrm{o}\mathrm{t}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}},$

and, according to the caption of their Figure 4, $E_0$ is "… the energy of a nonrotating sphere of equal mass and volume." Drawing from our separate discussion of the Maclaurin spheroid sequence, it would be reasonable to assume that the energy normalization adopted by MPT77 is the same as the normalization used by [T78], namely,

$E_{\mathrm{T}\mathrm{7}\mathrm{8}}$

$=$

$(\frac{4\pi }{3}{)}^{1/3}G(M^5\rho {)}^{1/3}.$

For models along the Maclaurin spheroid sequence, this normalization leads to expressions for the two key energy terms of the form,

$\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}}{E_{\mathrm{T}\mathrm{7}\mathrm{8}}}$	$=$	$\frac{3}{2\cdot 5}[(3-2e^2)\frac{{\sin }^{-1}e}{e}-3(1-e^2{)}^{1/2}]\frac{(1-e^2{)}^{1/6}}{e^2},$
$\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{T}\mathrm{7}\mathrm{8}}}$	$=$	$-\frac{3}{5}(1-e^2{)}^{1/6}\cdot \frac{{\sin }^{-1}e}{e};$

in which case, in the limit of a nonrotating sphere,

$\underset{e\to 0}{\lim }\left[\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{T}\mathrm{7}\mathrm{8}}}\right]$

$=$

$-\frac{3}{5}.$

But in Figure 4 of MPT77, the point along the Maclaurin spheroid sequence — the solid, black curve — that represents a nonrotating $(L_{*}=0)$ sphere has a normalized energy, $(E_{\mathrm{t}\mathrm{o}\mathrm{t}}/E_0)=-1.$ We conclude, therefore, that the normalization adopted by MPT77 is,

$E_0$

$=$

$\frac{3}{5}{E}_{\mathrm{T}\mathrm{7}\mathrm{8}}.$

Our Figure 2 (immediately above) attempts to quantitatively replicate the behavior of the Maclaurin spheroid sequence that is shown in Figure 4 (p. 213) of MPT77; the ordinate depicts, on a base-10 logarithmic scale, how the total energy varies with the spheroid's angular momentum over the range, $0\le L_{*}\le 1.50$, where the ordinate is,

$\frac{E_{\mathrm{t}\mathrm{o}\mathrm{t}}}{E_0}$	$=$	$\frac{5}{3}[\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}}{E_{\mathrm{T}\mathrm{7}\mathrm{8}}}+\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{T}\mathrm{7}\mathrm{8}}}]$
	$=$	$\frac{1}{2}[(3-2e^2)\frac{{\sin }^{-1}e}{e}-3(1-e^2{)}^{1/2}]\frac{(1-e^2{)}^{1/6}}{e^2}-(1-e^2{)}^{1/6}\cdot \frac{{\sin }^{-1}e}{e}$
	$=$	$\frac{(1-e^2{)}^{1/6}}{2e^2}\left\{\right[(3-2e^2)\frac{{\sin }^{-1}e}{e}-3(1-e^2{)}^{1/2}]-\frac{2e^2{\sin }^{-1}e}{e}\}$
	$=$	$\frac{(1-e^2{)}^{1/6}}{2e^2}[(3-4e^2)\frac{{\sin }^{-1}e}{e}-3(1-e^2{)}^{1/2}].$

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) — hereafter, EH85 — 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; the numbers in the first four columns have been drawn directly from Table 1 of EH85. The quantity, $E_{\mathrm{E}\mathrm{H}\mathrm{8}\mathrm{5}}$, that has been used to normalize the total energy in, for example, the fourth column of this table, is given by the expression,

$E_{\mathrm{E}\mathrm{H}\mathrm{8}\mathrm{5}}$	$\equiv$	$(4\pi G{)}^2{M}^5/J^2.$
📚 Eriguchi & Hachisu (1985), §2.2, p. 291, Eq. (7)

For purposes of comparison between the separate published works of MPT77 and EH85, here we desire to shift back to the normalization adopted by MPT77, namely,

Now, in a separate discussion of energy normalizations,

Data extracted from Table 1 (p. 290) of … Y. Eriguchi & I. Hachisu (1985) Maclaurin Hamburger Sequence Astronomy and Astrophysics, Vol. 148, pp. 289 - 292				Our Determination
$h_0^2$	$j^2$	$\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}}{|W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}|}$	$\frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{E}\mathrm{H}\mathrm{8}\mathrm{5}}}$	$L_{*}\equiv (4\pi /3{)}^{2/3}(3j^2{)}^{1/2}$
$5.802\times 10^{-2}$	$3.964\times 10^{-2}$	$0.445$	$-1.03\times 10^{-3}$	$0.8961$
$5.834\times 10^{-2}$	$3.816\times 10^{-2}$	$0.439$	$-1.02\times 10^{-3}$	$0.8792$
$5.916\times 10^{-2}$	$3.752\times 10^{-2}$	$0.437$	$-1.02\times 10^{-3}$	$0.8718$
$6.075\times 10^{-2}$	$3.718\times 10^{-2}$	$0.436$	$-1.02\times 10^{-3}$	$0.8678$
$6.416\times 10^{-2}$	$3.209\times 10^{-2}$	$0.412$	$-9.73\times 10^{-4}$	$0.8063$
$6.766\times 10^{-2}$	$3.090\times 10^{-2}$	$0.403$	$-9.61\times 10^{-4}$	$0.7912$
$7.070\times 10^{-2}$	$3.016\times 10^{-2}$	$0.389$	$-9.53\times 10^{-4}$	$0.7816$
$6.739\times 10^{-2}$	$3.192\times 10^{-2}$	$0.376$	$-9.74\times 10^{-4}$	$0.8041$
$5.376\times 10^{-2}$	$3.751\times 10^{-2}$	$0.368$	$-1.04\times 10^{-3}$	$0.8717$
$4.007\times 10^{-2}$	$4.502\times 10^{-2}$	$0.365$	$-1.12\times 10^{-3}$	$0.9550$
$3.202\times 10^{-2}$	$5.100\times 10^{-2}$	$0.366$	$-1.18\times 10^{-3}$	$1.0164$
$2.525\times 10^{-2}$	$5.975\times 10^{-2}$	$0.369$	$-1.26\times 10^{-3}$	$1.1002$
$1.811\times 10^{-2}$	$7.364\times 10^{-2}$	$0.378$	$-1.37\times 10^{-3}$	$1.2214$
$1.127\times 10^{-2}$	$9.914\times 10^{-2}$	$0.396$	$-1.53\times 10^{-3}$	$1.4171$
$4.812\times 10^{-3}$	$1.618\times 10^{-1}$	$0.430$	$-1.76\times 10^{-3}$	$1.8104$

See Also