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In Fowler's work, it is assumed that the system is completely degenerate (T=0). The case of a partially degenerate self-gravitating Fermi gas at arbitrary temperature has been discussed more recently by Hertel and Thirring (1971) and Chavanis (2002) in the context of statistical mechanics. They describe the phase transition, below a critical temperature Tc, from a gaseous configuration to a condensed state (with a degenerate core surrounded by a halo). This provides a simple physical mechanism showing how the system can reach highly degenerate configurations as a result of gravitational collapse; see Chavanis (2006) for a review on phase transitions in self-gravitating systems.
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Landau (1932) made, independently, an equivalent calculation and argued that for M>Mc quantum mechanics cannot prevent the system from collapsing to a point. However, he did not take this collapse very seriously and noted that, in reality, there exists stars with mass M>Mc that do not show this "ridiculous tendency," so that they must possess regions in which the laws of quantum mechanics are violated.
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There exists a formal analogy between the Chandrasekhar limiting mass MChandra=5.76M/μ2 for relativistic white dwarf stars in D=3 (corresponding to a polytrope n=n3=3), the critical mass Mc=8π of bacterial populations described by the Keller-Segel model in D=2 (corresponding to n=n3=+), and the critical mass Mc=4kBT/Gm or critical temperature kBTc=GMm/4 of self-gravitating isothermal systems in D=2 (corresponding to n=n3=+). At these critical values M=Mc or T=Tc, the system forms a Dirac peak. This analogy, sketched in Sec. 8 of astro-ph/0604012v1, is addressed specifically in.
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We could simply account for dissipative effects by introducing a friction force -ξu in the Euler Eq. 2. In that case Ẇ=-ξ ρu2dr≤0, so that the system evolves so as to minimize the functional W[ρ,u] at fixed mass.
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We could simply account for dissipative effects by introducing a friction force -ξu in the Euler Eq. 2. In that case Ẇ=-ξ ρu2dr≤0, so that the system evolves so as to minimize the functional W[ρ,u] at fixed mass.
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