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M. C. Miller
nuclear asymmetry to fit neutron star data. This is not fatal, because it means that
we are simply comparing model predictions with data, but it does mean that P (ρ)
cannot be inferred blindly without models.
If additional assumptions are made, for example that between fiducial densities
the equation of state is a polytrope P ∝ ρ γ , then [166] have shown that precise
mass and radius measurements of as few as three neutron stars could suffice to
give an empirically determined P (ρ). The inferred P (ρ) would then be compared
with the predictions from microphysical equations of state. As discussed by [131],
there are additional relatively model-independent constraints on the equation of
state that one can infer from observations. For example, the typical radius of a
neutron star scales as the quarter power of the pressure near nuclear saturation
density, and the maximum density that can be reached in a neutron star is ρ max ≈
1.5 × 10 16 g cm −3 (M /M max ) 2 if the maximum mass is M max .
Our final note about the theoretical predictions is that there are some phenomena
that have little effect on the mass-radius relation but are important for other
observables. For example, the existence of a proton superconducting gap can modify
core cooling dramatically [172], but the predicted gap energies of ∼0.1 − 1 MeV
[172] are so small compared to the Fermi energy that the overall structure of
neutron stars will be affected minimally. Thus if neutron star temperatures and
ages, particularly those of isolated neutron stars, can be inferred reliably, then they
will provide a beautiful complement to the mass and radius measurements that are
emphasized more in this review.
1.3 Constraints on Mass from Binary Observations
Mass measurements of neutron stars in binaries provide the most certain of all
constraints on the properties of cold high-density matter, particularly when the
companion to the neutron star is also a neutron star and thus the system approaches
the ideal of two point masses. In this section we discuss such measurements,
beginning with what can be learned from purely Newtonian observations and
moving on to the greater precision and breaking of degeneracies that are enabled
by measurements of post-Keplerian parameters from systems involving pulsars.
1.3.1 Newtonian Observations of Binaries
The classical approach to mass measurements in binaries assumes that one sees
periodic variation in the energy of spectral lines from one of the stars in the binary,
which we will call star 1. The period of variation is the orbital period P orb , the shape
of the variation gives the eccentricity e of the orbit, and the magnitude K 1 (which
has dimensions of speed) of the variation indicates the line-of-sight component of
the orbital speed of star 1. Using Kepler’s laws these observed quantities can be
M. C. Miller
nuclear asymmetry to fit neutron star data. This is not fatal, because it means that
we are simply comparing model predictions with data, but it does mean that P (ρ)
cannot be inferred blindly without models.
If additional assumptions are made, for example that between fiducial densities
the equation of state is a polytrope P ∝ ρ γ , then [166] have shown that precise
mass and radius measurements of as few as three neutron stars could suffice to
give an empirically determined P (ρ). The inferred P (ρ) would then be compared
with the predictions from microphysical equations of state. As discussed by [131],
there are additional relatively model-independent constraints on the equation of
state that one can infer from observations. For example, the typical radius of a
neutron star scales as the quarter power of the pressure near nuclear saturation
density, and the maximum density that can be reached in a neutron star is ρ max ≈
1.5 × 10 16 g cm −3 (M /M max ) 2 if the maximum mass is M max .
Our final note about the theoretical predictions is that there are some phenomena
that have little effect on the mass-radius relation but are important for other
observables. For example, the existence of a proton superconducting gap can modify
core cooling dramatically [172], but the predicted gap energies of ∼0.1 − 1 MeV
[172] are so small compared to the Fermi energy that the overall structure of
neutron stars will be affected minimally. Thus if neutron star temperatures and
ages, particularly those of isolated neutron stars, can be inferred reliably, then they
will provide a beautiful complement to the mass and radius measurements that are
emphasized more in this review.
1.3 Constraints on Mass from Binary Observations
Mass measurements of neutron stars in binaries provide the most certain of all
constraints on the properties of cold high-density matter, particularly when the
companion to the neutron star is also a neutron star and thus the system approaches
the ideal of two point masses. In this section we discuss such measurements,
beginning with what can be learned from purely Newtonian observations and
moving on to the greater precision and breaking of degeneracies that are enabled
by measurements of post-Keplerian parameters from systems involving pulsars.
1.3.1 Newtonian Observations of Binaries
The classical approach to mass measurements in binaries assumes that one sees
periodic variation in the energy of spectral lines from one of the stars in the binary,
which we will call star 1. The period of variation is the orbital period P orb , the shape
of the variation gives the eccentricity e of the orbit, and the magnitude K 1 (which
has dimensions of speed) of the variation indicates the line-of-sight component of
the orbital speed of star 1. Using Kepler’s laws these observed quantities can be