Revision as of 19:56, 13 May 2016

Characteristics of Unstable Eigenvectors in Self-Gravitating Tori

Background

Imamura & Hadley Collaboration

Based especially on the analysis provided in Paper I and Paper II of the Imamura & Hadley collaboration, the eigenvectors that have drawn our attention thus far can be categorized as J-modes (as discussed, for example, in §3.2.1, Table 1 & Figure 2 of Paper II) or I-modes. Our initial attempts to construct fits to these eigenvectors empirically have been described in a separate chapter. We begin, here, a more quantitative analysis of the structure of these unstable eigenvectors by borrowing from a subsection of that chapter a table of equilibrium parameter values for the typical P- and E-mode models described in Table 2 of Paper II.

Table 1: P- and E-mode Model Parameters Highlighted in Paper II K. Z. Hadley, P. Fernandez, J. N. Imamura, E. Keever, R. Tumblin, & W. Dumas (2014, Astrophysics and Space Science, 353, 191-222)
Extracted from Table 2 or Table 4 of Paper II						Extracted from Fig. 3 or Fig. 4 of Paper II
Name	<math>~M_*/M_d</math>	<math>~(n, q)</math>^†	<math>~R_-/R_+</math>	<math>~r_\mathrm{outer} \equiv \frac{R_+}{R_\mathrm{max}}</math>	<math>~R_\mathrm{max}</math>	Eigenfunction
E1	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.101</math>	<math>~5.52</math>	<math>~0.00613</math>
E2	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.202</math>	<math>~2.99</math>	<math>~0.0229</math>
E3	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.402</math>	<math>~1.74</math>	<math>~0.159</math>
P1	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.452</math>	<math>~1.60</math>	<math>~0.254</math>
P2	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.500</math>	<math>~1.49</math>	<math>~0.403</math>
P3	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.600</math>	<math>~1.33</math>	<math>~1.09</math>
P4	<math>~100</math>	<math>~(\tfrac{3}{2}, 2)</math>	<math>~0.700</math>	<math>~1.21</math>	<math>~3.37</math>
^†In all three papers from the Imamura & Hadley collaboration, <math>~q = 2</math> means, "uniform specific angular momentum."

We ask, first, "Can we understand why the radial eigenfunction of, for example, model E2 — re-displayed here, on the right — exhibits a series of sharp dips whose spacing gets progressively smaller and smaller as the outer edge of the torus is approached?"

Radial Modes in Homogeneous Spheres

Before attempting to analyze natural modes of oscillation in a polytropic torus, it is useful to review what is known about radial oscillations in the geometrically simpler, uniform-density sphere. As we have reviewed in a separate chapter, Sterne (1937) was the first to recognize that the set of eigenvectors that describe radial modes of oscillation in a homogeneous, self-gravitating sphere can be determined analytically. In the present context, it is advantageous for us to pull Sterne's solution from a discussion of the same problem presented by Rosseland (1964). As we have summarized in a separate chapter, the relevant eigenvalue problem is defined by the following one-dimensional, 2^nd-order ODE:

<math>~ ( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2} + \frac{2}{\chi_0}\biggl[ 1 - 2\chi_0^2 \biggr] \frac{\partial \xi}{\partial \chi_0} + \biggl(4 - \frac{2}{\chi_0^2} \biggr) \xi</math>

<math>~ - \mathfrak{F} \xi \, , </math>

where, it is understood that the expression for the (spatially and temporally varying) radial location of each spherical shell is,

<math>~r(r_0,t) = r_0 + \delta r(r_0)e^{i\sigma t} \, ,</math>

and in the present context we are adopting the variable notation,

<math>~\chi_0</math>	<math>~\equiv</math>	<math>~\frac{r_0}{R} \, ,</math>
<math>~\xi</math>	<math>~\equiv</math>	<math>~\frac{\delta r}{R} \, ,</math>

and <math>~R</math> is the initial (unperturbed) radius of the sphere. Here, the eigenvalue is related to the physical properties of the homogeneous sphere via the relation,

<math>~\mathfrak{F} \equiv \frac{2}{\gamma_\mathrm{g}} \biggl[\biggl(\frac{3\sigma^2}{4\pi G\rho_0}\biggr) + (4 - 3\gamma_\mathrm{g}) \biggr] \, .</math>

Drawing from Rosseland's (1964) presentation — see his p. 29, and our related discussion — the following table details the eigenvectors (radial eigenfunction and associated eigenfrequency) for the three lowest radial modes (m = 2, 4, 6) that satisfy this wave equation; the figure displayed in the right-most column has been extracted directly from p. 29 of Rosseland and shows the behavior of the lowest five radial modes (m = 2, 4, 6, 8, 10, as labeled) over the interval <math>~0 \le \chi_0 \le 1</math>.

Table 2 Rosseland's (1964) Eigenfunctions for Homogeneous Sphere Figure in the right-most column extracted from p. 29 of Rosseland (1964) "The Pulsation Theory of Variable Stars" (New York: Dover Publications, Inc.)
Mode	Eigenfunction	Square of Eigenfrequency: <math>~3\sigma^2/(4\pi G\rho)</math>
<math>~m</math>	As Published	<math>~\frac{m}{2}(m+1)\gamma - 4</math>
<math>~2</math>	<math>~\xi = -2\chi_0</math>	<math>~3\gamma - 4</math>
<math>~4</math>	<math>~\xi = -\frac{20}{3}\chi_0 + \frac{28}{3}\chi_0^3</math>	<math>~10\gamma-4</math>
<math>~6</math>	<math>~\xi = -14\chi_0 + \frac{252}{5} \chi_0^3 - \frac{198}{5} \chi_0^5</math>	<math>~21\gamma-4</math>

We should point out that, except for the lowest (m = 2) mode, each of the radial eigenfunctions crosses zero at least once at some location(s) that resides between the center <math>~(\chi_0=0)</math> of and the surface <math>~(\chi_0 = 1)</math> of the sphere. More specifically, for each mode, <math>~m</math>, the number of such radial "nodes" is <math>~(m-2)/2</math>. The locations of these nodes is apparent from even a casual inspection of the figure presented in the right-most column of Table 2.

When these radial modes of oscillation are discussed in the astrophysics literature, the conditions that give rise to a dynamical instability are often emphasized. Specifically, each mode becomes unstable when <math>~\sigma^2</math> becomes negative, which translates into a value of <math>~\gamma < \gamma_\mathrm{crit}(m)</math> — see our related discussion of the properties of eigenfunction solutions in the context of Sterne's (1937) analysis of this stability problem. Here we want to emphasize that all of natural modes of oscillation exist even when the configuration is dynamically stable.

Singular Sturm-Liouville Problem

As we have discussed in a separate chapter, there is a class of eigenvalue problems in the mathematical physics literature that is of the "Singular Sturm-Liouville" type. These problems are governed by a one-dimensional, 2^nd-order ODE of the form,

<math>~ \lambda \Upsilon(x) </math>	<math>~=</math>	<math>~-(1-x)^{-\alpha}(1+x)^{-\beta} \cdot \frac{d}{dx} \biggl[ (1-x)^{\alpha+1}(1+x)^{\beta+1} \cdot \frac{d\Upsilon(x)}{dx} \biggr] </math>
	<math>~=</math>	<math>~ (x^2-1)\frac{d^2\Upsilon(x)}{dx^2} + [\alpha - \beta + (\alpha+\beta+2)x]\frac{d\Upsilon(x)}{dx} \, , </math>

where the pair of exponent values, <math>~(\alpha, \beta) </math>, is set by the specific physical problem, while the eigenfunction, <math>~\Upsilon(x)</math>, and associated eigenfrequency, <math>~\lambda</math>, are to be determined. For any choice of the pair of exponents, there is an infinite number <math>~(j = 0 \rightarrow \infty)</math> of analytically known eigenvectors that satisfy this governing ODE; they are referred to as Jacobi Polynomials. Specifically, the j^th eigenfunction is,

<math>~\Upsilon_j(x) = J_j^{\alpha,\beta}(x)</math>

<math>~\equiv</math>

<math>~(1-x)^{-\alpha}(1+x)^{-\beta} \biggl\{ \frac{(-1)^j}{2^j j!} \cdot \frac{d^j}{dx^j}\biggl[ (1-x)^{j+\alpha}(1+x)^{j+\beta} \biggr] \biggr\} \, ;</math>

and the corresponding j^th eigenvalue is,

<math>~\lambda_j^{\alpha,\beta}</math>

<math>~j(j+\alpha+\beta + 1) \, .</math>

Table 3 presents the first three eigenfunctions (j = 0, 1, 2), along with each corresponding eigenfrequency; the figure displayed in the right-most column has been extracted directly from §22 (p. 773) of Abramowitz and Stegun's (1964) Handbook of Mathematical Functions (tenth printing, December 1972, with corrections) and shows the behavior of the lowest five Jacobi polynomials (j = 1, 2, 3, 4, 5, as labeled) over the interval <math>~-1 \le x \le +1</math> and for the specific case of <math>~(\alpha,\beta) = (1.5, - 0.5)</math>.

Table 3: Example Eigenvector Solutions to the Singular Sturm-Liouville Problem
<math>~j</math>	<math>~J_j^{\alpha,\beta}(x)</math>	<math>~\lambda_j^{\alpha,\beta}</math>
<math>~0</math>	<math>~1</math>	<math>~0</math>
<math>~1</math>	<math>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math>	<math>~(\alpha+\beta+2)</math>
<math>~2</math>	<math>~ \tfrac{1}{8}(12+7\alpha + \alpha^2 + 7\beta+\beta^2+ 2\alpha\beta ) x^2 + \tfrac{1}{4}(3\alpha + \alpha^2 - 3\beta-\beta^2) x + \tfrac{1}{8}(-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta) </math>	<math>~2(\alpha+\beta+3)</math>
See also, eqs. (35)-(37) of Wolfram MathWorld; and §22 (p. 773) of Abramowitz and Stegun's (1964) Handbook of Mathematical Functions, from which the illustration on the right has been extracted.

Here we highlight the qualitative similarities between the behavior of the set of Jacobi polynomials [figure extracted from Abramowitz and Stegun (1964) and reprinted here in Table 3] and the set of eigenfunctions that describe normal modes of oscillation in homogeneous, self-gravitating spheres [figure extracted from Rosseland (1964) and reprinted here in Table 2]. As we shall presently see, Jacobi polynomials play an important role in illuminating the structure of normal modes of oscillation in Papaloizou-Pringle tori.

Papaloizou-Pringle Tori

Here we briefly summarize the work (published by several groups in the mid-1980s) that has been done to determine the properties of both stable and unstable, nonaxisymmetric oscillation modes in Papaloizou-Pringle tori. In an accompanying chapter we explain in detail how the two-dimensional, 2^nd-order PDE that governs the behavior of these modes was derived by each research group.

Papaloizou & Pringle (1985)

Performing their linear stability analysis in cylindrical coordinates, Papaloizou & Pringle (1985) considered temporal and spatial variations in, for example, the pressure of the form,

<math>~p(\varpi,\varphi,z,t) = p_0(\varpi,z) + p^'(\varpi,z)e^{i(m\varphi + \sigma t)} \, .</math>

Rather than focusing on <math>~p^'</math>, however, they chose to build the governing PDE around an enthalpy-like perturbation defined as,

<math>~W(\varpi,z) = \frac{p^'}{\rho_0\bar\sigma} \, ,</math>

where, <math>~\bar\sigma \equiv (\sigma + m\Omega)</math>, and <math>~\Omega(\varpi)</math> is the fluid's circular orbital frequency in the initially axisymmetric, equilibrium torus. The governing PDE appears as equation (2.19) in Papaloizou & Pringle (1985); for convenience and clarity, that key equation has been extracted from their paper and displayed in the following framed image.

Equation (2.19) extracted without modification from p. 803 of Papaloizou & Pringle (1985)

"The dynamical stability of differentially rotating discs. II"

Monthly Notices of the Royal Astronomical Society, vol. 213, pp. 799-820 © Royal Astronomical Society

Papaloizou and Pringle (1985, MNRAS, 213, 799)

Note that, in this equation, <math>~\kappa(\varpi)</math> is the (radially dependent) epicyclic frequency in the torus, and <math>~h^'</math> is the radial derivative of the configuration's specific angular momentum. As has been realized by a number of groups — and as we have demonstrated in our accompanying detailed discussion — this governing PDE simplifies considerably when considering only PP tori that have uniform specific angular momentum because both <math>~\kappa^2</math> and <math>~h^'</math> are zero. Hence, also, <math>~D = \bar\sigma^2</math>. For such systems, the governing PDE is,

<math>~\frac{ {\bar\sigma}^2 \rho_0^2 W}{\Gamma p_0 } </math>

<math>~ - \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi \cdot \frac{\partial W}{\partial \varpi} \biggr] + \frac{\rho_0 m^2 W }{\varpi^2} - \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W}{\partial z} \biggr) \, . </math>

Fundamentally, this is the governing PDE that Blaes (1985) used in his examination of the stability of PP tori. But he chose to shift to dimensionless variables and to employ a different meridional-plane coordinate system in his analysis. In an accompanying discussion, we show step-by-step how this expression morphs into the governing PDE that serves as the focus of the Blaes85 analysis. In what follows, we provide a brief summary of this mathematical transformation.

Blaes (1985)

Setup

Fig. 1 extracted without modification from p. 554 of Blaes (1985)

"Oscillations of Slender Tori"

Monthly Notices of the Royal Astronomical Society,

vol. 216, pp. 553-563 © Royal Astronomical Society

As is illustrated in his Figure 1 — which we have reprinted for convenience here, on the right — Blaes (1985) shifted from cylindrical coordinates to a (dimensionless) polar-coordinate <math>~(x,\theta)</math> system whose origin sits at the pressure-maximum of the initial, unperturbed Papaloizou-Pringle torus, a distance, <math>~\varpi_0</math>, from the symmetry axis of the cylindrical coordinate system. Mapping between these two coordinate systems is accomplished via the relations,

<math>~x^2 = \biggl(1-\frac{\varpi}{\varpi_0}\biggr)^2 + \biggl(\frac{z}{\varpi_0}\biggr)^2</math>	and	<math>~\theta = \tan^{-1}\biggl[\frac{\zeta}{1-\chi}\biggr] \, ;</math>
or
<math>~\frac{\varpi}{\varpi_0} = 1 - x\cos\theta</math>	and	<math>~\frac{z}{\varpi_0} = x\sin\theta \, .</math>

Furthermore, he set <math>~\Gamma = (n+1)/n</math>, and rewrote the (initial, unperturbed) equilibrium pressure and density distributions in terms of the dimensionless enthalpy distribution in the PP torus, namely,

<math>~p_0 </math>	<math>~=</math>	<math>~p_\mathrm{max} f^{n+1}\, ,</math>
<math>~\rho_0 </math>	<math>~=</math>	<math>~\rho_\mathrm{max} f^{n}\, ,</math>

where, the two-dimensional dimensionless enthalpy distribution is,

<math>~f(x,\theta) </math>	<math>~=</math>	<math>~1 - \frac{x^2}{\beta^2}\biggl[ 1 + x(3\cos\theta -\cos^3\theta) \biggr] \, ,</math>
<math>~\beta^2</math>	<math>~\equiv</math>	<math>~\frac{2n}{\mathfrak{M}_0^2} \, ,</math>

and <math>~\mathfrak{M}_0</math> is the Mach number of the circular, azimuthal flow at the pressure and density maximum. It is important to appreciate that <math>~\beta</math> is a dimensionless parameter whose value dictates the relative thickness of the equilibrium torus; slim tori have <math>~\beta \ll 1</math>.

Finally, Blaes replaced the perturbation variable, <math>~W</math>, preferred by Papaloizou & Pringle (1985) with an equivalent but dimensionless perturbation variable,

<math>~\delta W</math>

<math>~\equiv \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{p_\mathrm{max}} \biggr]W \, ,</math>

where <math>~\Omega_0</math> is the angular frequency at the pressure and density maximum. [Actually, Blaes (1985) calls this dimensionless variable <math>~W</math>, rather than <math>~\delta W</math>, so care must be taken when published equations from these separate studies are compared.] After working carefully through these modifications — again, see our accompanying discussion for details — Blaes arrives at the governing PDE (his equation 3.2) that is highlighted in the following bordered box. Notice that, in this published expression, <math>~\nu \equiv \sigma/\Omega_0</math>, represents the azimuthal-mode eigenfrequency, normalized to the system's orbital frequency at the origin of the Blaes85 coordinate system.

Equation (3.2) extracted without modification from p. 558 of Blaes (1985)

"Oscillations of Slender Tori"

Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 © Royal Astronomical Society

In a direct analogy with Sterne's (1937) analysis of normal modes of oscillation in homogeneous spheres — discussed above — the ultimate objective here is to determine what two-dimensional eigenfunction(s), <math>~\delta W_j(x,\theta)</math>, and corresponding eigenfrequency(ies), <math>~\nu_j</math>, satisfy this governing PDE for arbitrarily thick/thin PP tori. In general, both the eigenfunction and corresponding eigenfrequency should be treated as complex functions/numbers. As we summarize, below, Blaes (1985) derived analytic expressions that provide one approximate solution for tori with small, but finite, values of <math>~\beta</math>. But, first, we will briefly review how he derived an entire spectrum of analytically specifiable normal modes in the limit of "slender tori," that is, tori for which <math>~\beta</math> is effectively zero.

Normal Modes in Slender Tori

Establishing the Simpler Eigenvalue Problem

In what he termed the "slender torus approximation," Blaes (1985) found it advantageous to introduce a function, <math>~\eta</math>, defined in terms of the equilibrium enthalpy distribution, <math>~f</math>, such that,

<math>~\eta^2 \equiv 1 - f</math>

<math>~\frac{x^2}{\beta^2}\biggl[ 1 + x(3\cos\theta -\cos^3\theta) \biggr] \, .</math>

One nice feature of this parameter is that, for all PP tori, its value varies from zero at the density maximum (also the origin of the Blaes85 polar coordinate system) to unity at the surface of the torus. Also, in the thin torus limit <math>~(\beta \ll 1)</math>,

<math>~\approx</math>

so <math>~\eta</math> can effectively be used as the independent radial coordinate in place of <math>~x</math>. As we demonstrate in detail in an accompanying discussion, in the Blaes85 "slender torus approximation," some of the terms in his equation (3.2) governing PDE dominate over others, facilitating simplification. The result — equation (1.6) in Blaes (1985), also displayed in the following bordered box — is a well-defined eigenvalue problem whose solution(s) provide approximate descriptions of normal mode(s) of oscillation in slender PP tori. Notice that Blaes attaches the superscript, (0), to denote eigenvector solutions to this governing PDE are only approximate solutions valid in the slender torus approximation.

Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of Blaes (1985)

"Oscillations of Slender Tori"

Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 © Royal Astronomical Society

Trial Eigenfunction

Following Blaes (1985) — see our accompanying discussion for details — If we assume that the eigenfunction is of the form,

<math>~\delta W^{(0)}</math>

<math>~\biggl[\eta^{|k|} \exp(ik\theta) \biggr] \Upsilon(\eta) \, ,</math>

we find that the function, <math>~\Upsilon(\eta)</math>, must satisfy the one-dimensional, 2^nd-order ODE,

<math>~(1-\eta^2) \frac{d^2\Upsilon}{d\eta^2} + \frac{1}{\eta}\biggl[(2|k|+1) - (2|k|+1+2n)\eta^2\biggr]\frac{d\Upsilon}{d\eta} </math>

<math>~ - 2n \biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon \, , </math>

which is a fairly standard looking eigenvalue problem. If, furthermore, we adopt an independent variable given by the expression,

<math>\Lambda \equiv 2\eta^2 - 1 \, ,</math>

in which case,

<math>\frac{d}{d\eta} \rightarrow 2^{3/2} (\Lambda+1 )^{1/2}\frac{d}{d\Lambda}</math> and <math>\frac{d^2}{d\eta^2} \rightarrow \biggl[ 4\frac{d}{d\Lambda} + 8(\Lambda+1)\frac{d^2}{d\Lambda^2} \biggr] \, , </math>

the statement of the eigenvalue problem becomes,

<math>~ \frac{n}{2} \biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math>

<math>~ (\Lambda^2-1)\frac{d^2\Upsilon}{d\Lambda^2} + \biggl[(n - |k|-1 ) + (|k|+1+n)\Lambda\biggr] \frac{d\Upsilon}{d\Lambda} \, . </math>

This definition of the eigenvalue problem is exactly of the singular Sturm-Liouville form, as described above, from which we draw the following coefficient associations:

<math>~(\alpha - \beta) \leftrightarrow (n-|k|-1)</math> and <math>~(\alpha+\beta+2) \leftrightarrow (|k|+1+n) \, ,</math>

that is,

<math>~\alpha \leftrightarrow (n-1)</math> and <math>~\beta\leftrightarrow |k|\, .</math>

Hence the j^th solution to this eigenvalue problem is,

<math>~\Upsilon_j(\Lambda)</math>

<math>~J_j^{n-1,|k|}(\Lambda) \, ,</math>

with the associated eigenvalue,

<math>~\frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)_j^2 -\|k\|\biggr] </math>	<math>~=</math>	<math>~j(j+n + \|k\|)</math>
<math>~\Rightarrow~~~ \biggl( \frac{\sigma}{\Omega_0} + m \biggr)_j^2 </math>	<math>~=</math>	<math>~\frac{1}{n}\biggl[2j^2+2jn + 2j\|k\| + n\|k\|\biggr] \, .</math>

Table 4 presents the polynomial expressions that are relevant to the first three (j = 0, 1, 2) eigenfunctions of "slender" PP-tori, along with the associated eigenvalues.

Table 4: Example Jacobi Polynomials Relevant to "Slender" PP-Tori Eigenvectors
<math>~j</math>	<math>~J_j^{n-1,\|k\|}(2\eta^2-1)</math>	<math>~\biggl( \frac{\sigma}{\Omega_0} + m \biggr)_j^2</math>
<math>~0</math>	<math>~1</math>	<math>~0</math>
<math>~1</math>	<math>~(n + 1 + \|k\|)\eta^2 - (1 + \|k\|)</math>	<math>~\tfrac{1}{n}[2+2n+(2+n)\|k\|]</math>
<math>~2</math>	<math>~ \tfrac{1}{2}[(6 + 5\|k\|+k^2) +n(5+2\|k\|) +n^2 ]\eta^4 - [4 + 4\|k\|+k^2 +n(2+\|k\|)]\eta^2 + \tfrac{1}{2} [2 + 3\|k\| + k^2] </math>	<math>~\tfrac{1}{n}[8+4n + (4+n)\|k\|] </math>

Analytic Solution

Piecing this together — including, as well, the time and azimuthal mode, m, dependence — we therefore ultimately conclude that,

<math>~\delta W_{j,k,m}^{(0)}</math>

<math>~\biggl[\eta^{|k|} \exp(ik\theta + m\varphi + \sigma t) \biggr] ~J_j^{n-1,|k|}(2\eta^2-1) \, ,</math>

and,

<math>~\biggl( \frac{\sigma}{\Omega_0} \biggr)_{j,k,m} </math>

<math>~-m \pm \biggl[\frac{2j^2+2jn + 2j|k| + n|k|}{n}\biggr]^{1/2} \, .</math>

This is a fantastic result, as it provides a totally analytic description of the eigenvectors that define a full spectrum <math>~(j = 0 \rightarrow \infty)</math> of normal-mode oscillations in slender tori that have uniform specific angular momentum and a range of reasonable polytropic indexes. As Blaes (1985) states — see the discussion immediately following his equation (1.8) — "The three parameters j (a non-negative integer), k (an integer) and m (an integer) completely describe the solution."

We need to keep in mind, however, that some restrictions accompany the slender torus approximation.

Each eigenvector represents a solution of an eigenvalue problem that is simpler than the eigenvalue problem defined by equation (2.19) of Papaloizou & Pringle (1985) — see above — or, equivalently, equation (3.2) of Blaes (1985) — see above.
The slender torus has a meridional cross-section that is a perfect circle whose cross-sectional radius is obtained by setting <math>~\eta = 1 ~~\Rightarrow~~r_\mathrm{torus} = \beta\varpi_0</math>.

And the analytically specified eigenvector exhibits the following simplified attributes:

As pointed out by Blaes (1985) — see the comments immediately following his equation (1.8) — for all values of <math>~(j,k,m)</math>, the eigenfrequency, <math>~\sigma/\Omega_0</math>, is real, hence all of the identified oscillation modes are stable.
For all values of <math>~(j,k)</math>, the eigenfunction is real and, as a result, the constant phase locus of each eigenvector will exhibit no azimuthal structure; see more discussion of this attribute, below.

Plots of a Few Example Eigenvectors

Blaes (1985) makes the following brief comments regarding the role of the indices, j and k, in defining the geometric distortions of the toroidal fluid configuration associated with each normal mode. "The significance of k is obvious — it is simply the number of wavelengths on a given isobaric surface [i.e., varying <math>~\theta</math> while holding <math>~\eta</math> constant] so that there are 2k nodal lines radiating from the torus centre at <math>~\eta = 0</math>." Alternatively, moving radially through the torus at a fixed <math>~\theta</math>, "In addition to the [radial] node at <math>~\eta = 0</math> which occurs for <math>~k \ne 0</math>, there are nodal surfaces at j other distinct values of <math>~\eta</math> inside the torus."

The plots presented as panels A, B, & D in Figure 1 display, for nine separate "slender torus" normal modes of oscillation, the radial variation of: (Middle) <math>~\delta W_{j,k}^{(0)}</math> versus <math>~\eta</math>; (Left) the associated Jacobi polynomial, <math>~J_{j}^{n-1,|k|}</math> versus <math>~x</math> where <math>~x = (2\eta^2-1)</math>; and (Right) <math>\log_{10}|\delta W_{j,k}^{(0)}|</math> versus <math>~\eta</math>. In panel A, all three displayed modes have j = 0; in panel B, all modes have j = 1; and in panel D, all modes have j = 2. In all three of these panels (A, B, D), blue curves are associated with k = 0; red curves are associated with k = 1; and green curves are associated with k = 2.

Figure 1: Blaes85 Eigenfunctions for Slender Tori with <math>~(n,q) = (\tfrac{3}{2},2)</math>
Panel A: j = 0 and (k = 0, 1, 2)

Panel B: j = 1 and (k = 0, 1, 2)

Panel C: Model E3 from Figure 4 of Hadley, et al. (2014)

Panel D: j = 2 and (k = 0, 1, 2)

Panel E: Model E2 from Figure 4 of Hadley, et al. (2014)

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 =====Plots of a Few Example Eigenvectors=====
+[http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] makes the following brief comments regarding the role of the indices, ''j'' and ''k'', in defining the geometric distortions of the toroidal fluid configuration associated with each normal mode.  "<font color="green">The significance of ''k'' is obvious &#8212; it is simply the number of wavelengths on a given isobaric surface</font> [''i.e.'', varying <math>~\theta</math> while holding <math>~\eta</math> constant] <font color="green">so that there are 2''k'' nodal lines radiating from the torus centre at</font> <math>~\eta = 0</math>."  Alternatively, moving radially through the torus at a fixed <math>~\theta</math>, "<font color="green">In addition to the</font> [radial] <font color="green">node at <math>~\eta = 0</math> which occurs for <math>~k \ne 0</math>, there are nodal surfaces at ''j'' other distinct values of <math>~\eta</math> inside the torus</font>."
 The plots presented as panels A, B, &amp; D in Figure 1 display, for nine separate "slender torus" normal modes of oscillation, the radial variation of:  (Middle) <math>~\delta W_{j,k}^{(0)}</math> versus <math>~\eta</math>; (Left) the associated Jacobi polynomial, <math>~J_{j}^{n-1,|k|}</math> versus <math>~x</math> where <math>~x = (2\eta^2-1)</math>; and (Right) <math>\log_{10}|\delta W_{j,k}^{(0)}|</math> versus <math>~\eta</math>.  In panel A, all three displayed modes have j = 0; in panel B, all modes have j = 1; and in panel D, all modes have j = 2.  In all three of these panels (A, B, D), blue curves are associated with k = 0; red curves are associated with k = 1; and green curves are associated with k = 2.

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Contents

Characteristics of Unstable Eigenvectors in Self-Gravitating Tori

Background

Imamura & Hadley Collaboration

Radial Modes in Homogeneous Spheres

Singular Sturm-Liouville Problem

Papaloizou-Pringle Tori

Papaloizou & Pringle (1985)

Blaes (1985)

Setup

Normal Modes in Slender Tori

Establishing the Simpler Eigenvalue Problem

Trial Eigenfunction

Analytic Solution

Plots of a Few Example Eigenvectors

See Also

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