Stability of spherical models in MOND

Figure 1.  We calculated numerically the solution $ \rho_s $ of (41) with central value $ \rho_s(0) = s $. In the above figure we plotted the mass $ M_s $ of $ \rho_s $ against the central value $ s\in[0.01, 1] $. We see that in the depicted range for every given mass $ M>0 $ there is only one $ s $ such that $ \rho_s $ has mass $ M $. The behaviour of the graph of $ M_s $ for $ s $ very small (the deep MOND limit) and for $ s $ very large (the Newtonian limit) is given in Figure 2

Figure 2.  The two diagrams show the same information as Figure 1 but with different ranges. In the left diagram, we zoomed in on the range $ s\in[0.0001, 0.01] $. This is the deep MOND limit. In this range the graph of $ M_s $ is approximately a parabola. For comparison we plotted (dashed line) also the parabola $ 1.06\, s^2 $. In the right diagram, we zoomed out on the range $ s\in[10, 1000] $. This is the Newtonian limit. There the graph of $ M_s $ becomes a straight line. Putting the informations from Figure 1 and Figure 2 together, we conclude that for every given mass $ M>0 $ there is exactly one $ s>0 $ such that $ \rho_s $ has mass $ M $

Figure 3.  Choice of $ \alpha_0 $