5.2 The structure of spherically-symmetric steady states

All solutions described so far have the property that the support of ρ contains a ball about the centre. In [138] Rein showed that steady states also exist whose support is a finite, spherically-symmetric shell with a vacuum region in the center. In [8Jump To The Next Citation Point] it was shown that there are shell solutions, which have an arbitrarily thin thickness. A systematic study of the structure of spherically-symmetric static solutions was carried out mainly by numerical means in [22Jump To The Next Citation Point] and we now present the conclusions of this investigation.

By prescribing the value μ(0), the equations can be solved, but the resulting solution will in general not satisfy the boundary condition μ(∞ ) = 0, but it will have some finite limit. It is then possible to shift both the cut-off energy E0 and the solution by this limit to obtain a solution, which satisfies μ (∞ ) = 0. A convenient way to handle the problem that E0 and μ(0) cannot both be treated as free parameters is to use the ansatz

f(x,v) = (1 − E ∕E0)k+ (L − L0)l+, k ≥ 0,l > − 1 ∕2,k < 3l + 7∕2,

as in [22Jump To The Next Citation Point]. This gives an equation for μ, which can be rewritten in terms of the function

eμ(r) y(r) = -E--. 0

In this way the cut-off energy disappears as a free parameter of the problem and we thus have the four free parameters k,l,L0 and y(0). The structure of the static solutions obtained in [22Jump To The Next Citation Point] is as follows:

If L0 = 0 the energy density can be strictly positive or vanish at r = 0 (depending on l) but it is always strictly positive sufficiently close to r = 0. Hence, the support of the matter is an interval [0,R ] 1 with R > 0 1, and we call such states ball configurations. If L0 > 0 the support is in an interval [R0, R1],R0 > 0, and we call such steady states for shells.

The value y(0) determines how compact or relativistic the steady state is, and the smaller values the more relativistic. For large values, recall y(0) ≤ 1∕E0, a pure shell or a pure ball configuration is obtained, cf. Figure 1View Image for a pure shell. Note that we depict the behavior of ρ but we remark that the pressure terms behave similarly but the amplitudes of p and pT can be very different, i.e., the steady states can be highly anisotropic.

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Figure 1: A pure shell

For moderate values of y(0) the solutions have a distinct inner peak and a tail-like outer peak, and by making y(0) smaller more peaks appear, cf. Figure 2View Image for the case of ball configurations.

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Figure 2: Multi-peaks of ball configurations, L0 = 0

In the case of shells there is a similar structure but in this case the peaks can either be separated by vacuum regions or by thin atmospheric regions as in the case of ball configurations. An example with multi-peaks, where some of the peaks are separated by vacuum regions, is given in Figure 3View Image.

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Figure 3: Multi-peaks of a shell

A different feature of the structure of static solutions is the issue of spirals. For a fixed ansatz of the density function f, there is a one-parameter family of static solutions, which are parameterized by y(0). A natural question to ask is how the ADM mass M and the radius of the support R change along such a family. By plotting for each y (0 ) the resulting values for R and M a curve is obtained, which reflects how radius and mass are related along such a one-parameter family of steady states. This curve has a spiral form, cf. Figure 4View Image. It is shown in [22Jump To The Next Citation Point] that in the isotropic case, where l = L0 = 0 the radius-mass curves always have a spiral form.

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Figure 4: (R, M ) spiral for k = 0, l = 10.5, L0 = 0, 0.01 ≤ y(0) ≤ 0.99

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