1 Astrophysical Constraints on Dense Matter in Neutron Stars
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T F = E F /k ≈ 10 12 K, which is much hotter than the expected interior temperatures
T < 10 10 K typical of neutron stars more than a few years old [172]. Neutron stars
are strongly degenerate.
Note, however, that the high Fermi energy of neutrons suggests the possibility
of additional particles at high densities. For example, the lambda particle has a rest
mass of m Λ = 1115.6 MeV/c 2 , so if the neutron Fermi energy exceeds 176 MeV
then the lambda is in principle stable because 176 MeV plus the neutron restmass energy 939.6 MeV exceeds 1115.6 MeV. Several other particles are within
300 MeV/c 2 of the neutron. In addition, because the density at the center of a neutron
star is a few times nuclear saturation density ρ nuc ≈ 2.6 × 10 14 g cm −3 , quarks
may become deconfined or matter might transition to a state that is lower-energy
than nucleonic matter even at zero pressure (strange matter; see, e.g., [77]). Various
density thresholds are summarized in Fig. 1.1.
Stars supported by nonrelativistic degeneracy pressure (a reasonable approximation for neutron stars, because E F < m n c 2 ) have radii that decrease with increasing
mass in contrast to most other objects. To see this, consider a star with a mass M
and radius R supported by nonrelativistic fermions of mass m. The Fermi energy per
particle is E F ∼ p 2
F /2m ∼ ¯
h
2 n 2/3 ∼ (M/R 3 ) 2/3 ∼ M 2/3 /R 2 . The gravitational
Fig. 1.1 Total energy per free neutron versus mass density (solid line). Above ∼10 13 g cm −3 the
Fermi energy starts to contribute palpably to the total, and above ∼10 15 g cm −3 the total energy
can exceed the rest mass energy of particles such as Λ 0 , Σ + , Δ, and Ξ 0 (marked by horizontal
dotted lines). Interactions between these particles can change the threshold density. The central
densities of realistic neutron stars range from ∼5 × 10 14 g cm −3 to ∼ few × 10 15 g cm −3 , so some
of these exotic particles may indeed be energetically favorable. Also marked are the densities at
which free electrons become relativistic; where those electrons have enough total energy to make
p + e − → n + ν e possible; where free neutrons can exist stably (i.e., at neutron drip); nuclear
saturation density ρ nuc ; and where free neutrons have a Fermi energy equal to their rest-mass
energy. To calculate the neutron Fermi energy we assume that all the mass is in free neutrons;
in reality at least a few percent of the mass is in protons and other particles, and below ρ nuc a
significant fraction of mass is in nuclei
5
T F = E F /k ≈ 10 12 K, which is much hotter than the expected interior temperatures
T < 10 10 K typical of neutron stars more than a few years old [172]. Neutron stars
are strongly degenerate.
Note, however, that the high Fermi energy of neutrons suggests the possibility
of additional particles at high densities. For example, the lambda particle has a rest
mass of m Λ = 1115.6 MeV/c 2 , so if the neutron Fermi energy exceeds 176 MeV
then the lambda is in principle stable because 176 MeV plus the neutron restmass energy 939.6 MeV exceeds 1115.6 MeV. Several other particles are within
300 MeV/c 2 of the neutron. In addition, because the density at the center of a neutron
star is a few times nuclear saturation density ρ nuc ≈ 2.6 × 10 14 g cm −3 , quarks
may become deconfined or matter might transition to a state that is lower-energy
than nucleonic matter even at zero pressure (strange matter; see, e.g., [77]). Various
density thresholds are summarized in Fig. 1.1.
Stars supported by nonrelativistic degeneracy pressure (a reasonable approximation for neutron stars, because E F < m n c 2 ) have radii that decrease with increasing
mass in contrast to most other objects. To see this, consider a star with a mass M
and radius R supported by nonrelativistic fermions of mass m. The Fermi energy per
particle is E F ∼ p 2
F /2m ∼ ¯
h
2 n 2/3 ∼ (M/R 3 ) 2/3 ∼ M 2/3 /R 2 . The gravitational
Fig. 1.1 Total energy per free neutron versus mass density (solid line). Above ∼10 13 g cm −3 the
Fermi energy starts to contribute palpably to the total, and above ∼10 15 g cm −3 the total energy
can exceed the rest mass energy of particles such as Λ 0 , Σ + , Δ, and Ξ 0 (marked by horizontal
dotted lines). Interactions between these particles can change the threshold density. The central
densities of realistic neutron stars range from ∼5 × 10 14 g cm −3 to ∼ few × 10 15 g cm −3 , so some
of these exotic particles may indeed be energetically favorable. Also marked are the densities at
which free electrons become relativistic; where those electrons have enough total energy to make
p + e − → n + ν e possible; where free neutrons can exist stably (i.e., at neutron drip); nuclear
saturation density ρ nuc ; and where free neutrons have a Fermi energy equal to their rest-mass
energy. To calculate the neutron Fermi energy we assume that all the mass is in free neutrons;
in reality at least a few percent of the mass is in protons and other particles, and below ρ nuc a
significant fraction of mass is in nuclei