LSUsimulations

"Fission" Simulations at LSU

This chapter essentially replicates an earlier html-based discussion.

Simulations Performed by John Cazes

Here we summarize the results of numerical simulations that John Cazes performed as part of his doctoral dissertation research, which was completed in 1999. A portion of this work has been published by 📚 J. E. Cazes & J. E. Tohline (2000, ApJ, Vol 532, Issue 2, pp. 1051 - 1068) in an article titled, "Self-Gravitating Gaseous Bars.   I. Compressible Analogs of Riemann Ellipsoids with Supersonic Internal Flows."

Model A

Table 1 extracted from p. 1057 of … J. E. Cazes & J. E. Tohline (2000) Self-gravitating Gaseous Bars.   I. Compressible Analogs of Riemann Ellipsoids with Supersonic Internal Flows The Astrophysical Journal, Vol 532, Issue 2, pp. 1051 - 1068

The initial configuration of "Model A" is an axisymmetric,

$n=\frac{3}{2}$

polytrope with differential rotation prescribed in such a way that the angular momentum as a function of cylindrical radius,

$\varpi$

, has the same function as a uniformly rotating,

$n=0$

polytrope — that is, the same as a Maclaurin spheroid. This is a rotation profile originally introduced by 📚 R. Stoeckly (1965, ApJ, Vol. 142, pp. 208 - 228). With this in mind, it is clear that "Model A" sits on the so-called

$n^{\prime }=0$

sequence, as introduced by 📚 J. P. Ostriker & J. W.-K. Mark (1968, ApJ, Vol. 151, pp. 1075 - 1088), and as implemented, for example, by 📚 P. Bodenheimer & J. P. Ostriker (1973, ApJ, Vol. 180, pp. 159 - 170), and by 📚 B. K. Pickett, R. H. Durisen, & G. A. Davis (1996, ApJ, Vol. 458, pp. 714 - 738). (See our additional discussion of Simple Rotation Profiles.)

The initial values of various "Model A" parameters are provided in the second column of numbers in Table 1 (p. 1057) of 📚 Cazes & Tohline (2000) — this table has been reprinted here, on the right. For example, the ratio of the maximum density to the mean density of the model is, ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}/\overline{\rho }=5.66$; and the ratio of rotational kinetic energy to the absolute value of the gravitational potential energy, $T_{\mathrm{r}\mathrm{o}\mathrm{t}}/|W|=0.30$. For comparison — see the first column of numbers — in a nonrotating, spherically symmetric $n=\frac{3}{2}$ polytrope, ${\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}/\overline{\rho }=5.99$ and $T_{\mathrm{r}\mathrm{o}\mathrm{t}}/|W|=0.0$.

1^st Video

Our initially axisymmetric "Model A" configuration is dynamically unstable toward the development of a so-called bar-mode instability; actually, the unstable eigenfunction has a slight two-armed spiral character. The caption of the icon shown here, on the left, contains a link to a YouTube video that shows the nonlinear development of this instability.

Model A Barmode Instability YouTube video:  BhRUqZe0Ly4

Resolution …   $[128,128,128]$ in $[\varpi ,Z,\phi ]$

• Radial:   Model A was evolved on a cylindrical grid that contained 128 radial zones, but the equatorial radius of the initial axisymmetric model extended out to only zone 53. Hence, ample room was allowed for radial expansion of the model as it deformed into a two-armed spiral, then bar-like, configuration.
• Vertical:   Although the model was initially quite flat — having a pole-to-equatorial axis ratio, $c/a=0.208$, meaning that the vertical extent above its equatorial plane was substantially less than 53 vertical zones — it was evolved on a grid that contained 128 vertical zones in this "northern" hemisphere. (Our technique for solving the Poisson equation imposed this grid size.) We assumed reflection symmetry through the equatorial plane, so it was not necessary to include an additional 128 zones in the "southern" hemisphere.
• Azimuthal:   In the azimuthal direction, each grid zone had an angular extent of $2\pi /256$. However, by forcing the evolution to maintain a two-fold "π-symmetry" only 128 grid zones were required to evolve the system with this resolution.

This evolution is viewed from an inertial frame of reference in which the initial symmetry (vertical) axis is tipped somewhat toward the viewer. As a consequence, even though the initial model is axisymmetric, it appears to be slightly ellipsoidal in even the earliest frames of the video. After "ejecting" a small portion of its mass into a "circumstellar" disk/ring, the model settles down into a nearly steady-state, rotating bar-like configuration that, for all intents and purposes is a compressible analog of a Riemann ellipsoid. (This is the theme of the 📚 Cazes & Tohline (2000) publication.)

See Also