Phase space structure

We searched the parameter space, or phase space, labeling our solutions using the two dimensionless parametersof of the polytropic equatio of state (24) discussed earlier. This procedure is typical when we have dimensionless parameters in our model. They determine the physics of our solution space, because they are independent of the scale.

A perl script was written to survey the parameter space of $C$ and $\lambda$. The different phases are described as follows (Figure 2.4),

• Phase Peak (Blue) Here the mass solution goes to a maximum, that we call peak, and then goes down again to stabilize in some constant value for large $\rho$. The radius behave in a similar manner (Figure 2.4). The characteristic feature we found in this sector behavior is that the size of the star is non-zero as $\rho(0)$ decreases and approaches zero, i.e. the radius has a finite value even if the mass goes to zero in the same limit $\rho(0)\rightarrow 0$. Eventually both mass and radius go down to zero, and as $\lambda$ increases we see not large deviations. The largest stars seem to correspond to the most massive. (Figure 2.4).
• Phase Plateau (Red) In these regime the mass of the star stabilizes remarkably after climbing fast to a maximum value, then it turns down slowly to stay in a constant value for large $\rho(0)$. It forms some kind of plateau. The radius goes down to a minimum, and then recovers and turns up again for large $\rho(0)$..(Figure 2.4
• Phase Increase (Green) In this phase the mass seems to grow withour bound and the raddi decreases to turn up at some point as $\rho$ increases. These solutions seems to be the least interesting, just monotomic increase for the mass and decrease for the radius (Figure 2.4).

Figure 24: Top: Survey of the parameter space; bottom: the conjetured continuum phase space structure.

Figure 25: Peak Phase (Blue), in $C=1$, $\lambda =2.5$

Figure 26: Plateau Phase (Red), in $C=0.01$, $\lambda =0.6$

Figure 27: Increase Phase (Green), in $C=0.01$, $\lambda =0.6$