Numerical Study of Polytropes with n=1 and Differential Rotation

TL;DR

We investigate equilibrium configurations of self-gravitating, rotating polytropes with index $n=1$ under Newtonian gravity to understand how differential rotation shapes the structure of rotating stars and planets. Using the ROTAT code for full 2D equilibria and a complementary semi-analytic Helmholtz-type method, we map the global solution space as a function of the differential-rotation parameter and identify three configuration families—T, P, and M—with exotica appearing near moderate differential rotation and large fragmentation-parameter values. 3D FLASH simulations validate hydrostatic stability for selected exotic configurations over tens of revolutions, motivating future dynamical stability studies and analytic descriptions for strongly deformed regimes. The results reveal a rich, degenerate topology of rotating equilibria and provide a framework for connecting Newtonian models to more realistic, rapidly rotating astrophysical objects such as neutron stars and giant planets.

Abstract

The solution space of differentially rotating polytropes with n=1 has been studied numerically. The existence of three different types of configurations: from spheroids to thick tori, hockey puck-like bodies and spheroids surrounded by a torus, separate from or merging with the central body has been proved. It has been shown that the last two types appear only at moderate degrees of rotation differentiality, sigma~2. Rigid-body or weakly differential rotation, as well as strongly differential, have not led to any "exotic" types of configurations. Many calculated configurations have had extremely large values of parameter tau, which has raised the question of their stability with respect to fragmentation.

Figures (11)

• Figure 1: Mass of a star $M_{\mathrm{s}}$ (left) and its equatorial radius $R_{\mathrm{e}}$ (right) as functions of parameter $\theta$ for several $\sigma$ values. The arrows indicate the configurations whose structures are shown in the graphs below.
• Figure 2: Structure of configurations of $\mathbf{T}_1$ ($\sigma=1$) and $\mathbf{T}_2$ ($\sigma=1.8$) marked by arrows in Fig. \ref{['Pix:MRlow']} in normalized coordinates $\{\xi/R_{\mathrm{e}},z/R_{\mathrm{e}}\}$. Details are in the text.
• Figure 3: Structure of configurations $\mathbf{T}_3$ and $\mathbf{P}_1$ (see Fig. \ref{['Pix:MRlow']}).
• Figure 4: Structure of configurations $\mathbf{M}_{1,2}$ (see Fig. \ref{['Pix:MRlow']}).
• Figure 5: Dependencies $M_{\mathrm{s}}(\theta)$, $R_{\mathrm{e}}(\theta)$, $\tau(\theta)$, and $\omega(\tau)$. Configurations of different types are highlighted in color. The large symbols with markings show the configurations shown below in Figs. \ref{['Pix:Tor']}, \ref{['Pix:Puck']}, and \ref{['Pix:Matryoshka']}. Other details are in the text.