User:Tohline/Apps/Ostriker64
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Polytropic & Isothermal Tori
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Overview
Here we will focus on the analysis of the structure selfgravitating tori that are composed of compressible — specifically, polytropic and isothermal — fluids as presented in a series of papers by Jeremiah P. Ostriker:
 J. Ostriker (1964, ApJ, 140, 1056) — The Equilibrium of Polytropic and Isothermal Cylinders
 J. Ostriker (1964, ApJ, 140, 1067) — The Equilibrium of SelfGravitating Rings
 J. Ostriker (1964, ApJ, 140, 1529) — On the Oscillations and the Stability of a Homogeneous Compressible Cylinder
 J. Ostriker (1965, ApJ Supplements, 11, 167) — Cylindrical Emden and Associated Functions
I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of S. Chandrasekhar.
Coordinate System
Basics
In §IIa of Paper II, Ostriker defines a set of orthogonal coordinates, , that is related to the traditional Cartesian coordinate system, , via the relations,









As Ostriker states, "The coordinate is the distance from a reference circle of radius (later chosen to be the major radius of the ring) …" The angle, , plays the role of the azimuthal angle, as is familiar in both cylindrical and spherical coordinates, while, here, is a meridionalplane polar angle measured counterclockwise from the equatorial plane. For axisymmetric systems, there will be no dependence on the azimuthal angle, so the pair of relevant coordinates in the meridional plane are,



and, 



Figure 1 extracted without modification from p. 1077 of J. P. Ostriker (1964; Paper II)
"The Equilibrium of SelfGravitating Rings"
ApJ, vol. 140, pp. 10671087 © American Astronomical Society 
For later reference, we note that (see eq. 3 of Paper II) the corresponding line element is,



which means that the relevant scale factors for the adopted coordinate system, , are
and the relevant differential volume element is,



Relationship to Toroidal Coordinate
Referring back to our separate discussion of the basic elements of a toroidal coordinate system, we know that, the meridionalplane toroidal coordinates are related to traditional meridionalplane cylindrical coordinate pair via the expressions,



and, 



assuming that the cylindricalcoordinate location of the anchor ring is . Let's determine how to transform between these two sets of coordinate pairs.
Independent Exploration
First, eliminating reference to Ostriker's "polar angle" , we see that,









Then, eliminating reference to Ostriker's radial coordinate , we find,









Now let's try to derive the alternate transformation. We'll start by eliminating the "polar angle" in toroidal coordinates.






The same relation also implies that,






Together, then, we have,









Alternatively, in an attempt to eliminate , we have,









But, also,





















Now that I think about it, this is all a bit silly because from the basic elements of a toroidal coordinate system we already know how to shift from cylindrical to toroidal coordinates.
Back to Basics
Mapping the other direction [see equations 2.13  2.15 of Wong (1973) ], we have,






where,



and 



and has the same sign as . Now, given that Ostriker's coordinates are related to cylindrical coordinates via the expressions,



and 



we can write,






and,



Hence,












Summary


Second Attempt
Single Offset Circle
Now an offcenter circle whose major and minor radii are, respectively, , will be described by the expression,



where both and are held constant while mapping out the variation of with . If we acknowledge that, in general, , then we know how varies with via the relation,





















In order to align this expression with the terminology (and variable labels) that we use in the context of a toroidal coordinate system, we associate the radius of the anchor ring as , and we associate the major radius of each circular torus as . We therefore have,






and, the coordinates of points along the surface of the torus are provided by the expressions,






We have tested this pair of expressions using Excel and have successfully demonstrated that they do, indeed, trace out a circle of radius, , whose center is offset from the symmetry axis by a distance, .
Set of Circles Whose Offset Increases With Circle Diameter
A set of nested offcenter circles will be described by allowing , that is, by having the offset distance, , vary with the size of the circle, . The above prescription for the normalized "coordinate" will work for any prescribed function.
But a particular function is demanded if we want this derived prescription to represent the behavior of toroidal coordinates. In a toroidal coordinate system, a specification of the value of the "radial" coordinate, , automatically dictates the ratio ; but we are not at liberty to separately define the value of the difference, . Instead, we must enforce the toroidalcoordinate relation,






where we have adopted the shorthand notation, . Hence,



Now, in a toroidal coordinate system, there is a similar "radial" coordinate, , whose value varies with distance from the anchor ring of radius, . Its value depends on both and via the relation,
This means that,












which also means that,



Case of Small Offset
Another way to look at this issue is to go back to the expression,






and assume that, while still dependent on the radial coordinate, the dimensionless offset is small. That is, assume that,
In this case, we can write,



And differentiating both sides of the expression with respect to gives,



COMMENT by Tohline (15 August 2018): I'm not sure that this is leading where I had hoped. I am gearing up to draw a comparison between these last expressions and eq. (74) in Ostriker's (1964) Paper II.
Gravitational Potential
Potential of a Thin Hoop
In §IIb of his Paper II, Ostriker (1964) derives an expression for the gravitational potential of a torus in the Thin Ring approximation, beginning specifically with the integral form of the Poisson equation that is widely referred to in the astrophysics community as an expression for the,
Scalar Gravitational Potential 




[BT87], p. 31, Eq. (23) 
(Note: Consistent with the usage favored by his doctoral dissertation advisor in [EFE], throughout his collection of 1964 papers Ostriker adopts a different sign convention as well as a different variable name to represent the gravitational potential.) Employing Ostriker's adopted coordinate system, and recognizing that, "the distance between the point of integration and the point of observation " is,



Ostriker's (1964) Paper II, p. 1070, Eq. (21) 
this expression for the gravitational potential becomes,
















Ostriker's (1964) Paper II, p. 1070, Eq. (22) 
where,

and 

Ostriker's (1964) Paper II, p. 1070, Eq. (23) 
Mapping back to cylindrical coordinates, for the moment, we recognize that,
Acknowledging as well that the mass of Ostriker's "thin hoop" is, , his expression for the potential becomes,
where,
After adopting the variable association, , it is clear that Ostriker's derived expression is identical to the Key Equation that we have identified elsewhere as providing the,

Series Expansion
In the context of Ostriker's expression for the potential, we see that,






Hence, in the vicinity of the ring where and is a "small parameter," we can draw on the binomial theorem and write,









Note, in particular, that,








and, 



Next we recognize that the following series expansion for the complete elliptic integral of the first kind — written in terms of the small parameter, — appears, for example, as eq. (8.113.3) in the Fourth Edition of Gradshteyn & Ryzhik (1965):









[This series expansion — up through the term — appears as equation 24 in Ostriker's (1964) Paper II.] Put together, then, Ostriker's expression for the gravitational potential in the thin ring approximation becomes,















where, again, we have recognized that the mass of the thin hoop is, . Now,









and, given that,



we also have,















So our series expansion for Ostriker's "thin ring" potential becomes,


















Finally, dropping the explicit mention of all terms and smaller gives the series expansion formulation presented by Ostriker, namely,






Ostriker's (1964) Paper II, p. 1071, Eq. (25) 
The Dimensionless Radial Coordinate, ξ, and Smallness Parameter, β
As we have reviewed separately, when researchers in the astrophysics community discuss the structure of spherical polytropes, the
invariably arises, as it is the governing 2^{nd}order ODE whose solution, , defines the internal structure of spherically symmetric equlibrium configurations. Traditionally, as well, the dimensionless radial coordinate,
is defined in terms of , which is a natural length scale of the (spherical) problem. Equation (42) of Ostriker's (1964) Paper II provides the traditional definition of . It is therefore not surprising that, even though Ostriker's set of 1964 papers deal largely with the equilibrium and stability of ringlike configurations, he adopts a similar definition for the dimensionless radial coordinate; specifically, eq. (5) of Paper II states that,
But, of course, in the context of Ostriker's presentation, is not a spherical radial coordinate but is, rather, as defined above; and is of the same order as the minor, crosssectional radius of the torus.
In eq. (6) of Paper II, Ostriker also defines the dimensionless parameter,
where is associated with the major radius of the ring. Then he states that, "… since (by hypothesis), we may be sure that …"
With the definitions of these two dimensionless parameters in hand — and, more specifically, after appreciating that,
— we can follow Ostriker's lead and rewrite his derived expression for in the form,






Ostriker's (1964) Paper II, p. 1071, Eq. (26) 
See Also
 T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.) — Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ringlike Objects
 W.T. Kim & S. Moon (2016, ApJ, 829, article id. 45, 22 pp.) — Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings
 E. Y. Bannikova, V. G. Vakulik & V. M. Shulga (2011, MNRAS, 411, 557  564) — Gravitational Potential of a Homogeneous Circular Torus: a New Approach
 D. Petroff & S. Horatschek (2008, MNRAS, 389,156  172) — Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity
The following quotes have been taken from Petroff & Horatschek (2008):
§1: "The problem of the selfgravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincaré (1885a,b,c) and Dyson (1892, 1893). Each of them tackled the problem of an axially symmetric, homogeneous ring in equilibrium by expanding it about the thin ring limit. In particular, Dyson provided a solution to fourth order in the parameter , where provides a measure for the radius of the crosssection of the ring and the distance of the crosssection's centre of mass from the axis of rotation."
§7: "In their work on homogeneous rings, Poincaré and Kowalewsky, whose results disagreed to first order, both had made mistakes as Dyson has shown. His result to fourth order is also erroneous as we point out in Appendix B." 
 P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113  3198) — Phase Transitions in SelfGravitating Systems
 M. Ansorg, A. Kleinwächter & R. Meinel (2003, MNRAS, 339, 515) — Uniformly Rotating Axisymmetric Fluid Configurations Bifurcating from Highly Flattened Maclaurin Spheroids
 M. Lombardi & G. Bertin (2001, Astronomy & Astrophysics, 375, 1091  1099) — Boyle's Law and Gravitational Instability
 W. Kley (1996, MNRAS, 282, 234) — Maclurin Discs and Bifurcations to Rings
 J. W. Woodward, J. E. Tohline, & I. Hachisu (1994, ApJ, 420, 247  267) — The Stability of Thick, SelfGravitating Disks in Protostellar Systems
 I. Bonnell & P. Bastien (1991, ApJ, 374, 610  622) — The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation
 J. E. Tohline & I. Hachisu (1990, ApJ, 361, 394  407) — The Breakup of SelfGravitating Rings, Tori, and Thick Accretion Disks
 F. Schmitz (1988, Astronomy & Astrophysics, 200, 127  134) — Equilibrium Structures of Differentially Rotating SelfGravitating Gases
 P. Veugelen (1985, Astrophysics & Space Science, 109, 45  55) — Equilibrium Models of Differentially Rotating Polytropic Cylinders
 M. A. Abramowicz, A. Curir, A. SchwarzenbergCzerny, & R. E. Wilson (1984, MNRAS, 208, 279  291) — SelfGravity and the Global Structure of Accretion Discs
 P. Bastien (1983, Astronomy & Astrophysics, 119, 109  116) — Gravitational Collapse and Fragmentation of Isothermal, NonRotating, Cylindrical Clouds
 Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870  1875) — Another Equilibrium Sequence of SelfGravitating and Rotating Incompressible Fluid
 J. E. Tohline (1980, ApJ, 236, 160  171) — Ring Formation in Rotating Protostellar Clouds
 T. Fukushima, Y. Eriguchi, D. Sugimoto, & G. S. BisnovatyiKogan (1980, Progress of Theoretical Physics, 63, 1957  1970) — Concave Hamburger Equilibrium of Rotating Bodies
 J. Katz & D. LyndenBell (1978, MNRAS, 184, 709  712) — The Gravothermal Instability in Two Dimensions
 P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584 597) — Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid (includes torus with nonuniform rotation)
 Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an equilibrium incompressible configuration must rotate uniformly on cylinders (the famous "PoincaréWavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"
 C. J. Hansen, M. L. Aizenman, & R. L. Ross (1976, ApJ, 207, 736  744) — The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders
 C.Y. Wong (1974, ApJ, 190, 675  694) — Toroidal Figures of Equilibrium
 C.Y. Wong (1973, Annals of Physics, 77, 279  353) — Toroidal and Spherical Bubble Nuclei
 Gunnar Randers (1942, ApJ, 95, 88) — The Equilibrium and Stability of RingShaped 'barred SPIRALS'.
 William Duncan MacMillan (1958), The Theory of the Potential, New York: Dover Publications
 Oliver Dimon Kellogg (1929), Foundations of Potential Theory, Berlin: Verlag Von Julius Springer
 Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148154) — On the Dynamics of Revolving Fluids
 F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041  1106) — The Potential of an Anchor Ring. Part II.
 In this paper, Dyson derives the gravitational potential inside the ring mass distribution
 F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43  95) — The Potential of an Anchor Ring. Part I.
 In this paper, Dyson derives the gravitational potential exterior to the ring mass distribution
 S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37) — Zusätze und Bemerkungen zu Laplace's Untersuchung über die Gestalt der Saturnsringe
 Poincaré (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). — references copied from paper by Wong (1974)
© 2014  2019 by Joel E. Tohline 