# Properties of Homogeneous Ellipsoids (2)

In addition to pulling from §53 of Chandrasekhar's EFE, here, we lean heavily on the papers by M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586) (hereafter, WT83) and by D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995a, ApJ, 446, 472) (hereafter, Paper I).

## Sequence-Defining Dimensionless Parameters

A Riemann sequence of S-type ellipsoids is defined by the value of the dimensionless parameter,

 $~f$ $~\equiv$ $~\frac{\zeta}{\Omega} =$ constant,

[ EFE, §48, Eq. (31) ]
[ WT83, Eq. (5) ]
[ Paper I, Eq. (2.1) ]

where, $~\zeta$ is the system's vorticity as measured in a frame rotating with angular velocity, $~\Omega$. Alternatively, we can use the dimensionless parameter,

 $~x$ $~\equiv$ $~\biggl[\frac{ab}{a^2 + b^2} \biggr]f \, ,$

[ EFE, §48, Eq. (40) ]
[ Paper I, Eq. (2.2) ]

or,

 $~\Lambda$ $~\equiv$ $~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f = -\Omega x \, .$

[ WT83, Eq. (4) ]

## Conserved Quantities

Algebraic expressions for the conserved energy, $~E$, angular momentum, $~L$, and circulation, $~C$, are, respectively,

 $~E$ $~=$ $~\frac{1}{2}v^2 + \frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I$ $~\rightarrow$ $~\cancelto{0}{\frac{1}{2}v^2} + \frac{1}{2} [(a+bx)^2 + (b+ax)^2]\Omega^2 - 2I \, ,$

[ 1st expression — EFE, §53, Eq. (239) ]
[ 2nd expression — Paper I, Eq. (2.7) ]

where — see an accompanying discussion for the definitions of $~A_1$, $~A_2$, and $~A_3$,

 $~I$ $~=$ $~A_1a^2 + A_2b^2 + A_3c^2 \, ;$

[ 1st expression — EFE, §53, Eq. (239) ]
[ 2nd expression — Paper I, Eq. (2.8) ]

 $~\frac{5L}{M}$ $~=$ $~(a^2 + b^2)\Omega - 2ab\Lambda$ $~=$ $~ (a^2 + b^2 + 2abx)\Omega \, ;$

[ 1st expression — EFE, §53, Eq. (240) ]
[ 2nd expression — Paper I, Eq. (2.5) ]

 $~\frac{5C}{M}$ $~=$ $~(a^2 + b^2)\Lambda - 2ab\Omega$ $~=$ $~- [2ab + (a^2 + b^2)x ]\Omega \, .$

[ 1st expression — EFE, §53, Eq. (241) ]
[ 2nd expression — Paper I, Eq. (2.6) ]

If we rewrite the expression for the system's free energy in terms of $~L$ (and x) instead of $~\Omega$ (and x), we have,

 $~E$ $~=$ $~\frac{1}{2} \biggl(\frac{5L}{M}\biggr)^2 \frac{(a+bx)^2 + (b+ax)^2}{(a^2 + b^2 + 2abx)^2} - 2I \, ,$

[ Paper I, Eq. (3.4) ]

Note that, based on the units chosen in Paper I, $~M = 5$, and $~abc = 15/4$.

# Aside: Chandra's Notation

According to equation (107) in §21 of EFE, it appears as though,

 $~A_i - A_j$ $~=$ $~- (a_i^2 - a_j^2)A_{ij} \, .$

And, according to equation (105) in §21 of EFE, it appears as though,

 $~B_{ij}$ $~=$ $~A_j - a_i^2A_{ij} \, .$

So, for example,

 $~A_{12}$ $~=$ $~-\biggl[ \frac{A_1 - A_2}{a_1^2 - a_2^2} \biggr] \, ,$

and,

 $~B_{12}$ $~=$ $~A_2 + a_1^2\biggl[ \frac{A_1 - A_2}{a_1^2 - a_2^2} \biggr]$ $~=$ $~\frac{(a_1^2 - a_2^2)A_2 + a_1^2(A_1 - A_2)}{a_1^2 - a_2^2}$ $~=$ $~\frac{a_1^2A_1 - a_2^2A_2 }{a_1^2 - a_2^2} \, .$

# Free Energy Surface(s)

## Scope

Consider a self-gravitating ellipsoid having the following properties:

• Semi-axis lengths, $~(x,y,z)_\mathrm{surface} = (a,b,c)$, and corresponding volume, $~4\pi/(3abc)$  ; and consider only the situations $0 \le b/a \le 1$ and $0 \le c/a \le 1$  ;
• Total mass, $~M$  ;
• Uniform density, $~\rho = (3 M)/(4\pi abc)$  ;
• Figure is spinning about its c axis with angular velocity, $~\Omega$  ;
• Internal, steady-state flow exhibiting the following characteristics:
• No vertical (z) motion;
• Elliptical (x-y plane) streamlines everywhere having an ellipticity that matches that of the overall figure, that is, $~e = (1-b^2/a^2)^{1/2}$  ;
• The velocity components, $~v_x$ and $~v_y$, are linear in the coordinate and, overall, characterized by the magnitude of the vorticity, $~\zeta$  .

Such a configuration is uniquely specified by the choice of six key parameters:   $~a$, $~b$, $~c$, $~M$, $~\Omega$, and $~\zeta$  .

## Free Energy of Incompressible, Constant Mass Systems

We are interested, here, in examining how the free energy of such a system will vary as it is allowed to "evolve" as an incompressible fluid — i.e., holding $~\rho$ fixed — through different ellipsoidal shapes while conserving its total mass. Following Paper I, we choose to set $~M = 5$ — which removes mass from the list of unspecified key parameters — and we choose to set $~\rho = \pi^{-1}$, which is then reflected in a specification of the semi-axis, $~a$, in terms of the pair of dimensionless axis ratios, $~b/a$ and $~c/a$, namely,

 $~a^3$ $~=$ $~\frac{3Ma^2}{4\pi(bc)\rho} = \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1}\, .$

Moving forward, then, a unique ellipsoidal configuration is identified via the specification of four, rather than six, key parameters —   $~b/a$, $~c/a$, $~\Omega$, and $~x$   — and the free energy of that configuration is given by the expression,

 $~E\biggl(\frac{b}{a}, \frac{c}{a}, \Omega, x\biggr)$ $~=$ $~\frac{a^2}{2} \biggl[\biggl(1+\frac{b}{a} \cdot x\biggr)^2 + \biggl(\frac{b}{a}+x\biggr)^2\biggr]\Omega^2 - 2I$ $~=$ $~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl\{\frac{1}{2} \biggl[\biggl(1+\frac{b}{a} \cdot x\biggr)^2 + \biggl(\frac{b}{a}+x\biggr)^2\biggr]\Omega^2 - \frac{2I}{a^2}\biggr\} \, ,$

where,

 $~x$ $~\equiv$ $~\biggl[\frac{(b/a)}{1 + (b/a)^2} \biggr]\frac{\zeta}{\Omega} \, ,$ $~\frac{I}{a^2}$ $~=$ $~\biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr] \, ,$

and the functional behavior of the coefficients, $~A_1$, $~A_2$, and $~A_3$, are given by the expressions provided in an accompanying discussion.

Alternatively, replacing $~\Omega$ in favor of $~L$, we have,

 $~E\biggl(\frac{b}{a}, \frac{c}{a}, L, x\biggr)$ $~=$ $~\frac{L^2}{2a^2} \biggl[ \biggl(1+\frac{b}{a}\cdot x \biggr)^2 + \biggl(\frac{b}{a}+x \biggr)^2 \biggr] \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 + 2\biggl(\frac{b}{a}\biggr)x \biggr]^{-2} - 2I$ $~=$ $~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3} \biggl[ \biggl(1+\frac{b}{a}\cdot x \biggr)^2 + \biggl(\frac{b}{a}+x \biggr)^2 \biggr] \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 + 2\biggl(\frac{b}{a}\biggr)x \biggr]^{-2}$ $~- 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .$

### Conserve Only L

Let's fix the total angular momentum, $~L$, of a triaxial configuration and examine how the configuration's free energy varies as we allow it to contort through different triaxial shapes — that is, as its pair of axis ratios varies, always maintaining $~\tfrac{b}{a} < 1$ — and as we vary $~x$, which characterizes the fraction of angular momentum that is stored in internal spin versus overall figure rotation. The desired free-energy function, $~E(\tfrac{b}{a},\tfrac{c}{a}, x)|_L$, has just been defined, but visualizing its behavior is difficult because, in this situation, the free energy is a warped, three-dimensional surface draped across the four-dimensional domain, $~(\tfrac{b}{a},\tfrac{c}{a}, x, E_L)$.

Acknowledging that we are primarily interested in identifying extrema of this free-energy function, the discussion presented in §3.2 of Paper I shows us how to reduce the dimensionality of this problem by one. There, it is shown that, as long as $~\tfrac{b}{a} \ne 1$, extrema exist in the $~x$-coordinate direction — that is, $~\partial E_L/\partial x = 0$ — only if $~x = 0.$ For a given choice of $~L$, therefore, the relevant two-dimensional free-energy surface is defined by the expression,

 $~E\biggl(\frac{b}{a}, \frac{c}{a}, x=0\biggr)\biggr|_L$ $~=$ $~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3} \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2\biggr]^{-1}$ $~ - 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .$

Figure 3 of Paper I presents a black-and-white contour plot of this $~E_L$ function for the specific case of $~L = 4.71488$, which, for reference, is the total angular momentum of an equilibrium Maclaurin spheroid having an eccentricity, $~e = 0.85$ (see Table 1, below). We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) Paper I publication and have reprinted it as the left-hand panel of our Figure 1. Note that we have flipped the plot horizontally and rotated it by 90° so that the orientation of the axis pair, $~(\tfrac{b}{a},\tfrac{c}{a})$, conforms with the orientation of a related, information-rich diagram presented by Chandrasekhar (1965) — see also our accompanying discussion of equilibrium sequence plots.

Figure 1: Free-Energy Surface Projected onto the $~(\tfrac{b}{a},\tfrac{c}{a})$ Plane
 All three contour plots show how the free-energy, $~E_L$, varies across the $~(\tfrac{b}{a}, \tfrac{c}{a})$ domain for the specific case of $~L = 4.71488$. Horizontal axis is $~0 \le \tfrac{b}{a} \le 1$ and vertical axis is $~0 \le \tfrac{c}{a} \le 1$. Left-hand Panel:Black-and-white contour plot extracted from p. 477 of Paper I "Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points" ApJ, vol. 446, pp. 472-484 © AAS Middle Panel:Black-and-white contour plot digitally overlaid on color contour plot. Right-hand Panel:Color contour plotcreated here as a projection of the free-energy surface shown in Fig. 2.

In our Figure 2, this same $~E_L$ function has been displayed as a warped, two-dimensional free-energy surface draped across the three-dimensional $~(\tfrac{b}{a},\tfrac{c}{a},E)$ domain, where depth as well as color has been used to tag energy values. The two-dimensional, colored contour plot presented in the right-hand panel of our Figure 1 results from the projection of this free-energy surface onto the $~(\tfrac{b}{a},\tfrac{c}{a})$ plane; it reproduces in quantitative detail the black-and-white contour plot that we have extracted from Paper I. In an effort to (qualitatively) illustrate this agreement, we have digitally "pasted" the black-and-white contour plot from Paper I onto our colored contour plot and presented the combined image in the middle panel of our Figure 1.

Our Figure 2 image of the free-energy surface helps illuminate the description of this surface that appears in the caption of Fig. 3 from Paper I. Quoting from that figure caption:  "The [equilibrium] Maclaurin spheroid sits on a saddle point $~[(\tfrac{b}{a},\tfrac{c}{a}) = (1.0,0.52678); E_0 = -7.81842]$, while a global minimum with $~E_0 = -7.83300$ exists at $~(\tfrac{b}{a},\tfrac{c}{a}) = (0.588,0.428)$."

Figure 2: Free-Energy Surface

### Animation

The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the $~(\tfrac{b}{a},\tfrac{c}{a})$ plane (left) for configurations having nineteen different total angular momentum values, $~L$, as detailed in column 5 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of $~L$. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge $~(\tfrac{b}{a} = 1)$ of the warped surface. For values of $~e < 0.81267$ — corresponding to $~L < 4.23296$ — the Maclaurin spheroid (marked by a small white circle/sphere) sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid (marked by a small dark-blue circle/sphere) is associated with a saddle point of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid (marked by a small white circle/sphere) having the same total angular momentum. As Paper I points out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.

Figure 3: Animation

 Table 1:  Parameter Values Associated with Each Frame of the Figure 3 Animation (parameter values associated with Figures 1 & 2 are highlighted in pink) Maclaurin Spheroid $~L^\dagger$ Jacobi Ellipsoids $~e$ $~\frac{c}{a}$ $~E_L$ $~E_\mathrm{plot}^\ddagger$ $~\frac{b}{a}$ $~\frac{c}{a}$ $~E_\mathrm{Jac}$ 0.650 0.7599342 -8.9018255 0.0 2.8270256 --- --- --- 0.675 0.7378177 -8.8165100 0.0 2.9985043 --- --- --- 0.700 0.7141428 -8.7216343 0.0 3.1820090 --- --- --- 0.725 0.6887489 -8.6155943 0.0 3.3796768 --- --- --- 0.750 0.66143783 -8.4963506 0.0 3.5942337 --- --- --- 0.775 0.6319612 -8.3612566 0.0 3.8292360 --- --- --- 0.790 0.6131068 -8.2711758 0.0 3.9819677 --- --- --- 0.795 0.6066094 -8.2394436 0.0 4.0351072 --- --- --- 0.800 0.6000000 -8.2067933 0.0 4.0894508 --- --- --- 0.805 0.5932748 -8.1731817 0.0 4.1450581 --- --- --- 0.810 0.5864299 -8.1385621 0.0 4.2019932 --- --- --- 0.815 0.5794610 -8.1028846 0.0064 4.2603252 0.880967 0.545588 -8.102934 0.820 0.5723635 -8.0660955 0.0524 4.3201286 0.797543 0.516311 -8.066596 0.825 0.5651327 -8.0281369 0.1116 4.3814839 0.744298 0.496028 -8.029578 0.830 0.5577634 -7.9889461 0.1665 4.4444785 0.702967 0.479341 -7.991848 0.835 0.5502499 -7.9484555 0.2140 4.5092074 0.668439 0.464724 -7.953367 0.840 0.5425864 -7.9065917 0.2551 4.5757737 0.638420 0.451485 -7.914095 0.845 0.5347663 -7.8632747 0.2912 4.6442903 0.611646 0.439241 -7.873990 0.850 0.5267827 -7.8184175 0.3232 4.7148806 0.587337 0.427750 -7.833003 †Here, the units of angular momentum are as used in Paper I. In order to convert to units of $~L$ as used in EFE (see, for example, Table I, in Chapter 5, §32), multiply by $~[2^2/(3\cdot 5^{10})]^{1/6} = 0.0717585$. ‡$~E_\mathrm{plot}$ is a normalized value of $~E_L$ that has been used for plotting purposes. It's definition is: $E_\mathrm{plot} = 0.25*\biggl\{ \log_{10}\biggl[0.0001 + \frac{(E_L + |E_\mathrm{Jac}|)}{|E_\mathrm{Jac}|} \biggr] + 4\biggr\}$
PROPERTIES OF VARIOUS MACLAURIN SPHEROIDS

eccenL       covera        omega2      ellChandra          L

0.815  5.7946096D-01  3.7625539D-01  3.0571438D-01  4.2603252D+00
0.820  5.7236352D-01  3.8058727D-01  3.1000578D-01  4.3201286D+00
0.825  5.6513273D-01  3.8489420D-01  3.1440854D-01  4.3814839D+00
0.830  5.5776339D-01  3.8917054D-01  3.1892894D-01  4.4444785D+00
0.835  5.5024994D-01  3.9341001D-01  3.2357378D-01  4.5092074D+00
0.840  5.4258640D-01  3.9760569D-01  3.2835048D-01  4.5757737D+00
0.845  5.3476630D-01  4.0174986D-01  3.3326712D-01  4.6442903D+00
0.850  5.2678269D-01  4.0583395D-01  3.3833257D-01  4.7148806D+00
0.855  5.1862800D-01  4.0984835D-01  3.4355656D-01  4.7876802D+00
0.860  5.1029403D-01  4.1378236D-01  3.4894980D-01  4.8628384D+00
0.865  5.0177186D-01  4.1762394D-01  3.5452411D-01  4.9405200D+00
0.870  4.9305172D-01  4.2135955D-01  3.6029264D-01  5.0209081D+00
0.875  4.8412292D-01  4.2497391D-01  3.6626999D-01  5.1042063D+00
0.880  4.7497368D-01  4.2844972D-01  3.7247248D-01  5.1906420D+00
0.885  4.6559102D-01  4.3176729D-01  3.7891846D-01  5.2804708D+00
0.890  4.5596052D-01  4.3490417D-01  3.8562861D-01  5.3739810D+00
0.895  4.4606614D-01  4.3783459D-01  3.9262639D-01  5.4714996D+00
0.900  4.3588989D-01  4.4052888D-01  3.9993856D-01  5.5733994D+00
0.905  4.2541157D-01  4.4295266D-01  4.0759585D-01  5.6801086D+00
0.910  4.1460825D-01  4.4506586D-01  4.1563375D-01  5.7921218D+00
0.915  4.0345384D-01  4.4682147D-01  4.2409362D-01  5.9100155D+00
0.920  3.9191836D-01  4.4816395D-01  4.3302405D-01  6.0344667D+00
0.925  3.7996710D-01  4.4902713D-01  4.4248265D-01  6.1662784D+00
0.930  3.6755952D-01  4.4933139D-01  4.5253852D-01  6.3064134D+00
0.935  3.5464771D-01  4.4897998D-01  4.6327550D-01  6.4560401D+00
0.940  3.4117444D-01  4.4785386D-01  4.7479681D-01  6.6165969D+00
0.945  3.2707033D-01  4.4580450D-01  4.8723156D-01  6.7898831D+00
0.950  3.1224990D-01  4.4264348D-01  5.0074442D-01  6.9781934D+00

PROPERTIES OF JACOBI ELLIPSOIDS THAT HAVE THE SAME ANGULAR MOMENTA (L) AS THE ABOVE MACLAURIN SPHEROIDS

e      b/a       c/a        A1        A2        A3       omega2       a        L_C        L      energy

0.815  0.880967  0.545588  0.474189  0.557354  0.968456  0.371826  1.983364  0.305714  4.260325 -8.102934
0.820  0.797543  0.516311  0.441622  0.589410  0.968968  0.366634  2.088279  0.310006  4.320129 -8.066596
0.825  0.744298  0.496028  0.419233  0.611283  0.969484  0.361394  2.165672  0.314409  4.381484 -8.029578
0.830  0.702967  0.479341  0.400927  0.629069  0.970004  0.356104  2.232633  0.318929  4.444479 -7.991848
0.835  0.668439  0.464724  0.384983  0.644489  0.970527  0.350761  2.293990  0.323574  4.509207 -7.953367
0.840  0.638420  0.451485  0.370620  0.658325  0.971055  0.345362  2.351945  0.328350  4.575774 -7.914095
0.845  0.611646  0.439241  0.357403  0.671009  0.971588  0.339905  2.407740  0.333267  4.644290 -7.873990
0.850  0.587337  0.427750  0.345063  0.682812  0.972126  0.334386  2.462170  0.338333  4.714881 -7.833003
0.855  0.564969  0.416851  0.333417  0.693915  0.972668  0.328802  2.515795  0.343557  4.787680 -7.791082
0.860  0.544173  0.406427  0.322334  0.704450  0.973216  0.323150  2.569038  0.348950  4.862838 -7.748172
0.865  0.524676  0.396390  0.311717  0.714514  0.973770  0.317425  2.622239  0.354524  4.940520 -7.704210
0.870  0.506269  0.386673  0.301490  0.724181  0.974329  0.311624  2.675686  0.360293  5.020908 -7.659127
0.875  0.488788  0.377221  0.291594  0.733511  0.974895  0.305741  2.729638  0.366270  5.104206 -7.612848
0.880  0.472100  0.367988  0.281979  0.742554  0.975467  0.299772  2.784332  0.372472  5.190642 -7.565289
0.885  0.456097  0.358937  0.272604  0.751348  0.976047  0.293710  2.840003  0.378918  5.280471 -7.516357
0.890  0.440687  0.350033  0.263435  0.759930  0.976635  0.287549  2.896880  0.385629  5.373981 -7.465947
0.895  0.425792  0.341249  0.254440  0.768329  0.977230  0.281283  2.955205  0.392626  5.471500 -7.413941
0.900  0.411344  0.332556  0.245593  0.776572  0.977835  0.274902  3.015230  0.399939  5.573399 -7.360207
0.905  0.397284  0.323929  0.236868  0.784683  0.978449  0.268398  3.077231  0.407596  5.680109 -7.304592
0.910  0.383556  0.315345  0.228242  0.792685  0.979073  0.261762  3.141512  0.415634  5.792122 -7.246923
0.915  0.370112  0.306780  0.219694  0.800598  0.979708  0.254980  3.208416  0.424094  5.910016 -7.187000
0.920  0.356903  0.298209  0.211202  0.808442  0.980356  0.248041  3.278337  0.433024  6.034467 -7.124589
0.925  0.343885  0.289608  0.202744  0.816240  0.981017  0.240927  3.351732  0.442483  6.166278 -7.059417
0.930  0.331013  0.280950  0.194298  0.824010  0.981692  0.233621  3.429144  0.452539  6.306413 -6.991160
0.935  0.318242  0.272206  0.185842  0.831774  0.982384  0.226102  3.511226  0.463276  6.456040 -6.919428
0.940  0.305523  0.263344  0.177348  0.839558  0.983094  0.218342  3.598775  0.474797  6.616597 -6.843746
0.945  0.292805  0.254325  0.168790  0.847385  0.983825  0.210310  3.692789  0.487232  6.789883 -6.763530
0.950  0.280029  0.245105  0.160133  0.855288  0.984579  0.201966  3.794537  0.500744  6.978193 -6.678040