1 Astrophysical Constraints on Dense Matter in Neutron Stars
17
The problem is that for several bursters, the most probable values of the observationally inferred quantities F TD,∞ , A, and D, combined with the model parameter
f c , yield α > 1/8. This would imply that the mass and radius are complex numbers.
For example, in their analysis of 4U 1820–30, [95] used a Gaussian prior probability
distribution for F TD,∞ , which had a mean F 0 = 5.39 × 10 −8 erg cm −2 s −1
and a standard deviation σ F = 0.12 × 10 −8 erg cm −2 s −1 . They also used a
Gaussian prior probability distribution for A, with A 0 = 91.98 (km/10 kpc) 2 and
σ A = 1.86 (km/10 kpc) 2 . Their prior probability distribution for D was a boxcar
distribution with a midpoint D 0 = 8.2 kpc and a half-width of ΔD = 1.4 kpc.
Finally, they assumed a boxcar prior probability distribution for the color factor,
with f c0 = 1.35 and a half-width Δf c = 0.05. If we take the midpoint of each
distribution and also follow [95] by assuming that the opacity is dominated by
Thomson scattering and thus κ = 0.2 cm 2 g −1 for the pure helium composition
appropriate to 4U 1820–30, we find α = 0.179 > 1/8.
Güver et al. [95] note that the probability of obtaining a viable M and R for
4U 1820–30 from these equations drops with increasing distance, but if we reduce
D to the 6.8 kpc, which is the smallest value allowed in the priors of [95], and
keep the other input parameters fixed, we find α = 0.148. If we also increase f c
to its maximum value of 1.4 and take the +2σ value of A and the −2σ value of
F TD,∞ , α is still 0.129. In fact [206] showed that if we consider the prior probability
distribution of F TD,∞ , A, D, and f c used by Güver et al. [95], only a fraction 1.5 ×
10 −8 of that distribution yields real numbers for M and R. This demonstrates that
the 4% fractional uncertainties on the mass and radius of this neutron star obtained
by Güver et al. [95] emerge from the theoretical assumptions rather than from the
data. Thus such apparent precision is actually a red flag that one or more of the
model assumptions is incorrect.
The first suggestion for which assumption is in error came from [206], who
proposed that although the entire surface still emits uniformly throughout the
cooling phase, the photospheric radius might be larger than the radius of the star, i.e.,
r ph > R. However, analysis of the cooling phase of the superburst from 4U 1820–
30 [158] demonstrates that such a solution is disallowed for at least the superburst
emission from this star, because any detectable change in photospheric radius would
require a flux very close to Eddington, and such fluxes give extremely poor fits to
the data. The work of [158] used and verified the fully relativistic Comptonized
spectral models of [211], and also showed that the fraction of the surface that
emits changes systematically throughout the superburst (the emitting area drops
by ∼20% during the ∼1600 s analyzed). Moreover, there is no guarantee that the
whole surface was emitting at any time. Thus the star does not emit uniformly
over its entire surface during the superburst, and hence it cannot be assumed that
it has uniform emission during shorter bursts when the data quality is insufficient
to check this assumption. Indeed, the presence of burst oscillations (see [230] for
a recent review) demonstrates that there is nonuniformity in burning during many
bursts. Additional concerns are that the color factor is likely to evolve during the
burst, and that some of the bursts are not fit well using existing spectral models
[43, 55, 87, 89, 210, 242].
17
The problem is that for several bursters, the most probable values of the observationally inferred quantities F TD,∞ , A, and D, combined with the model parameter
f c , yield α > 1/8. This would imply that the mass and radius are complex numbers.
For example, in their analysis of 4U 1820–30, [95] used a Gaussian prior probability
distribution for F TD,∞ , which had a mean F 0 = 5.39 × 10 −8 erg cm −2 s −1
and a standard deviation σ F = 0.12 × 10 −8 erg cm −2 s −1 . They also used a
Gaussian prior probability distribution for A, with A 0 = 91.98 (km/10 kpc) 2 and
σ A = 1.86 (km/10 kpc) 2 . Their prior probability distribution for D was a boxcar
distribution with a midpoint D 0 = 8.2 kpc and a half-width of ΔD = 1.4 kpc.
Finally, they assumed a boxcar prior probability distribution for the color factor,
with f c0 = 1.35 and a half-width Δf c = 0.05. If we take the midpoint of each
distribution and also follow [95] by assuming that the opacity is dominated by
Thomson scattering and thus κ = 0.2 cm 2 g −1 for the pure helium composition
appropriate to 4U 1820–30, we find α = 0.179 > 1/8.
Güver et al. [95] note that the probability of obtaining a viable M and R for
4U 1820–30 from these equations drops with increasing distance, but if we reduce
D to the 6.8 kpc, which is the smallest value allowed in the priors of [95], and
keep the other input parameters fixed, we find α = 0.148. If we also increase f c
to its maximum value of 1.4 and take the +2σ value of A and the −2σ value of
F TD,∞ , α is still 0.129. In fact [206] showed that if we consider the prior probability
distribution of F TD,∞ , A, D, and f c used by Güver et al. [95], only a fraction 1.5 ×
10 −8 of that distribution yields real numbers for M and R. This demonstrates that
the 4% fractional uncertainties on the mass and radius of this neutron star obtained
by Güver et al. [95] emerge from the theoretical assumptions rather than from the
data. Thus such apparent precision is actually a red flag that one or more of the
model assumptions is incorrect.
The first suggestion for which assumption is in error came from [206], who
proposed that although the entire surface still emits uniformly throughout the
cooling phase, the photospheric radius might be larger than the radius of the star, i.e.,
r ph > R. However, analysis of the cooling phase of the superburst from 4U 1820–
30 [158] demonstrates that such a solution is disallowed for at least the superburst
emission from this star, because any detectable change in photospheric radius would
require a flux very close to Eddington, and such fluxes give extremely poor fits to
the data. The work of [158] used and verified the fully relativistic Comptonized
spectral models of [211], and also showed that the fraction of the surface that
emits changes systematically throughout the superburst (the emitting area drops
by ∼20% during the ∼1600 s analyzed). Moreover, there is no guarantee that the
whole surface was emitting at any time. Thus the star does not emit uniformly
over its entire surface during the superburst, and hence it cannot be assumed that
it has uniform emission during shorter bursts when the data quality is insufficient
to check this assumption. Indeed, the presence of burst oscillations (see [230] for
a recent review) demonstrates that there is nonuniformity in burning during many
bursts. Additional concerns are that the color factor is likely to evolve during the
burst, and that some of the bursts are not fit well using existing spectral models
[43, 55, 87, 89, 210, 242].