G io ii i2

Figure 3.1. Mass fractions of different constituents of the outer envelope of a newly born neutron star versus matter density in beta equilibrium at different temperatures Tg = T/ (109 K) (after Haensel et al. 1996). Calculations are performed for the Lattimer & Swesty (1991) model of nucleon matter with a specific choice (K0 = 220 MeV) of the incompressibility of cold symmetric nuclear matter at the saturation density.

We will summarize the results using a more recent version of the compressible liquid-drop model, formulated and developed by Lattimer & Swesty (1991), selecting a specific value of the incompressibility of symmetric nuclear matter at saturation (equilibrium) density, K0 = 220 MeV (for the definition of K0, see § 5.4). We assume nuclear equilibrium as well as beta equilibrium of the matter. The assumption of nuclear equilibrium is justified by high temperature. Beta equilibrium is adopted for simplicity; a very rapid cooling of matter at highest temperatures can produce deviations from beta equilibrium.

In Fig. 3.1 we show the composition of the hot matter of the neutron star envelope for T = 5 x 109 K, 8 x 109 K, and 1.2 x 1010 K. We restrict ourselves to p < 1013 g cm-3, because at higher densities the thermal effects on matter composition are negligible. At T > 5 x 109 K, the shell and pairing effects, so visible in the T = 0 (ground state) approximation, particularly through jumps in the density dependence of various quantities (see § 3.2), are washed out by the thermal effects.

At T = 1.2 x 1010K, the nuclei evaporate completely for p < 109 g cm-3. This can be understood within the compressible liquid-drop model; the nuclei are then considered as droplets of nuclear matter. At p < 1011 g cm-3, these droplets have to coexist with a vapor of neutrons, protons and a-particles. However, the coexistence of two different nucleon phases (denser - nuclear liquid, less dense - vapor of nucleons and a-particles) is possible only at T lower than some critical temperature at given a density, Tcrit(p). For p < 109 g cm-3, one has Tcrit(p) < 1.2 x 1010 K.

With decreasing temperature, the mass fraction of evaporated nucleons and a-particles decreases. For T = 8 x 109 K, a-particles are present at p <

1010 g cm-3, while free protons appear at even lower p. Free neutrons are present at all densities, but their fraction does not exceed one percent for p <

At T = 5 x 109 K the thermal effects are weak and imply mainly the appearance of a small fraction of free neutrons ("neutron vapor") below zero-temperature neutron drip density pND; this fraction falls below 10-5 at p = 1010 g cm-3. Further decrease of T leads to the disappearance of neutrons at p < pND, and to switching-on of shell effects. Another important effect will be the onset of superfluidity of neutrons (both inside and outside the nuclei) and protons. The composition freezes and does not change with further decrease of the temperature. An initially fluid element of the matter solidifies if its temperature falls below the melting temperature Tm that depends on local density and composition (§ 2.3.3).

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