Numerical Experiments

We performed a significan number of experiments in an attempt to understand the solutions of our equations. The first set of plots we have show a group of families of curves.

We ploted families of solutions labeled by $\lambda$ for $C=1$. With this type of plots we can understand the physics of the model in function of the central initial density $\rho(0)$, and the two dimensionless parameters $X$ and $\lambda$.

In figure (10 we have the total stellar mass and stellar radius in function of the central density, with $C=0.1$, each curve labeled by $\lambda$. As $\lambda$ decreases closing to $1.35$ the mass reached a maximum, so we think stars beyond that point are not stable. As the value of $\lambda$ increases another maximum appears. We also observe that as the curves approach $\rho=0$, the radius of the object decreases for $\lambda$ above $2$ (roughly), and increases below these value. The mass increases in direct proportion to $\lambda$ at large density, and at low density we see the opposite trend. The radius seems to be stable at large $\rho$, regardless $\lambda$.

In figure (11) we have simular curves for $C=1$. When the central density is large, the mass increases when $\lambda$ decreases. The radius follows a similar behavior, increasing as $\lambda$ decreases. The general characteristic of these families is that both the mass and the stellar radius increase in the limit $\rho\rightarrow 0$ above a certain critical value of $\lambda$ (which looks like $1.35$) and decreases below it.

The third set of families of curves shows the dependence on $C$ at $\lambda =3.0$ (Curves in figure 12). As $C$ decreases the maximum peak softens and almost dissapears for small values of $C$. The stellar mass and radius reach a maximum, decreases a bit, and stabilizes, in direct proportion of $\rho$ increases.

Using logarithmic plots is a commom practice in astrophysics, given the overlaping of scales that parameters and observables cover in these type of phenomena.

We investigated the solutions of out TOV equations in a systematic way for $C=1$ and several values of $\lambda$. The results are shown in the rest of the figures of this report. Subsecuently we investigated the parameter space of $C$ and $\lambda$ and generated a significant catalog of plots, using a perl script written by Pal. We were able to identify certain general features of the solutions in some approximate regimes..

Between $0$ and $\lambda =0.6$ we have solutions that describe objects with mass (Figure 2.3) that increases to a maximum, then decreases again. The radius goes down to a minimum, and then recovers and goes up.

Around $\lambda=1,33$ the stellar mass decreases, gets to a minimum and goes up again, growing and then turns again in a local maximum. The radii behaves in a similar manner. These phase stays the same up to around $\lambda=1.5$.

Starting from $\lambda =1.6$ up to $\lambda =2.0$ we see that the mass goes up to a maximum, then goes down again and stabilizes to a constant value, in direct proportion to the increase in $\rho(0)$. The radius starts with a certain finite value, different from zero, when $\rho(0)=0$, and then drops to some minimum value and then its not clear if turns up again, but there is evidence that it does.

Now in the region $2.0 < \lambda < 2.5$ the behavior is similar to the previos one. 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 $\rho(0)\rightarrow 0$.

For higer values of $\lambda$ we see not significant changes but now the radius seem to go to zero as $\rho(0)$ does (see fig 2.3 to 2.3).

Figure 10: Families of solutions labeled by $\lambda$ for $C=0.1$

Figure 11: Families of solutions labeled by $\lambda$ for $C=1$

Figure 12: Families of solutions labeled by $C$ for $\lambda =3$

Figure 13: $C=1$, $0.1< \lambda < 0.5$

Figure 14: $C=1$, $\lambda =1.33$

Figure 15: $C=1$, $\lambda =1.6$

Figure 16: $C=1$, $\lambda =1.8$

Figure 17: $C=1$, $\lambda =2.0$

Figure 18: $C=1$, $\lambda =2.3$

Figure 19: $C=1$, $\lambda =2.4$

Figure 20: $C=1$, $\lambda =2.5$

Figure 21: $C=1$, $\lambda =2.6$

Figure 22: $C=1$, $\lambda =3.0$

Figure 23: $C=1$, $\lambda =4.0$