18
M. C. Miller
Suleimanov et al. [210] find a radius of more than 14 km from a long PRE burst
from 4U 1724–307 based on fits of their spectral models to the decaying phase
of the burst. As part of their fits, they find that the Eddington flux occurs not
at touchdown, but at a 15% lower luminosity; this, therefore, calls into question
another of the assumptions in the standard approach to obtaining mass and radius
from bursts. Their radius value is based on the good fit they get to the bright portion
of the burst, when according to their fits the local surface flux exceeds ∼50%
of Eddington. This is an intriguing method that should be considered carefully
when data are available from the next generation of X-ray instruments. However,
a potential concern is that the spectra does not agree with their models below ∼50%
of Eddington. This suggests that there is other emission in the system at least at those
lower luminosities, and hence that some of this emission might be present at higher
luminosities as well. Against this is the excellent fit without extra components that
[158] obtained for the superburst from 4U 1820–30 using the models of [211]. More
and better data are the key.
If truly excellent spectra can be obtained, then as pointed out by [143], inference
of the surface gravity g and surface redshift z leads uniquely to determination of the
stellar mass and radius. However, even the ∼2 × 10 7 counts observed using RXTE
from the 4U 1820–30 superburst are insufficient to determine both g and z uniquely
[158], so this appears to require much larger collection areas. The combination (1 +
z)/g 2/9 can be measured precisely using sufficiently good continuum spectra, and
then combined with other measurements to, possibly, constrain M and R [138, 158],
so this is promising for the future. The net result is that currently inferred masses and
radii from spectral fits to thermonuclear X-ray bursts must be treated with caution;
none are reliable enough to factor into equation of state constraints.
1.4.2 Fits of Thermal Spectra to Cooling Neutron Stars
In principle, observations of cooling neutron stars with known distances allow us to
measure the radii of those stars, modulo an unknown redshift. In practice, as with
radius estimates from bursts, systematic errors dominate and thus current radius
determinations are not reliable enough to help constrain the properties of dense cold
matter.
To understand the basic principles, suppose that the star is at a distance d and that
we measure a detector bolometric flux F det,bol from the star that is fit by a spectrum
with an effective temperature T eff,∞ at infinity. Suppose that we also assume that
the entire surface radiates uniformly. If the surface redshift is z then the luminosity
at the surface is L surf = (1 + z) 2 L ∞ = (1 + z) 2 F det,bol 4πd 2 = 4πR 2 σ SB T 4
surf =
4πR 2 (1 + z) 4 σ SB T 4
eff,∞ . This implies
R = (1 + z)
−1 d[F det,bol /(σ SB T
4
eff,∞ )]
1/2 .
(1.13)
M. C. Miller
Suleimanov et al. [210] find a radius of more than 14 km from a long PRE burst
from 4U 1724–307 based on fits of their spectral models to the decaying phase
of the burst. As part of their fits, they find that the Eddington flux occurs not
at touchdown, but at a 15% lower luminosity; this, therefore, calls into question
another of the assumptions in the standard approach to obtaining mass and radius
from bursts. Their radius value is based on the good fit they get to the bright portion
of the burst, when according to their fits the local surface flux exceeds ∼50%
of Eddington. This is an intriguing method that should be considered carefully
when data are available from the next generation of X-ray instruments. However,
a potential concern is that the spectra does not agree with their models below ∼50%
of Eddington. This suggests that there is other emission in the system at least at those
lower luminosities, and hence that some of this emission might be present at higher
luminosities as well. Against this is the excellent fit without extra components that
[158] obtained for the superburst from 4U 1820–30 using the models of [211]. More
and better data are the key.
If truly excellent spectra can be obtained, then as pointed out by [143], inference
of the surface gravity g and surface redshift z leads uniquely to determination of the
stellar mass and radius. However, even the ∼2 × 10 7 counts observed using RXTE
from the 4U 1820–30 superburst are insufficient to determine both g and z uniquely
[158], so this appears to require much larger collection areas. The combination (1 +
z)/g 2/9 can be measured precisely using sufficiently good continuum spectra, and
then combined with other measurements to, possibly, constrain M and R [138, 158],
so this is promising for the future. The net result is that currently inferred masses and
radii from spectral fits to thermonuclear X-ray bursts must be treated with caution;
none are reliable enough to factor into equation of state constraints.
1.4.2 Fits of Thermal Spectra to Cooling Neutron Stars
In principle, observations of cooling neutron stars with known distances allow us to
measure the radii of those stars, modulo an unknown redshift. In practice, as with
radius estimates from bursts, systematic errors dominate and thus current radius
determinations are not reliable enough to help constrain the properties of dense cold
matter.
To understand the basic principles, suppose that the star is at a distance d and that
we measure a detector bolometric flux F det,bol from the star that is fit by a spectrum
with an effective temperature T eff,∞ at infinity. Suppose that we also assume that
the entire surface radiates uniformly. If the surface redshift is z then the luminosity
at the surface is L surf = (1 + z) 2 L ∞ = (1 + z) 2 F det,bol 4πd 2 = 4πR 2 σ SB T 4
surf =
4πR 2 (1 + z) 4 σ SB T 4
eff,∞ . This implies
R = (1 + z)
−1 d[F det,bol /(σ SB T
4
eff,∞ )]
1/2 .
(1.13)