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M. C. Miller
1.4.3 Modeling of Waveforms
Thermonuclear bursts that display brightness oscillations, and isolated millisecond
pulsars that are detectable in X-rays, have had their periodic waveforms analyzed
in attempts to constrain their masses and radii. The basic principle is simple: if one
makes the standard assumption that the oscillations are rotational modulations of the
flux produced by a hot spot fixed on the star, then the shape of the waveform encodes
information about the mass and radius (see [182], in which the relevant formulae
were derived and which was the first paper to use millisecond pulsar waveforms to
constrain the neutron star mass and radius). For example, a star of a given rotation
frequency that has a large radius will have a higher linear surface rotational speed
at a given latitude than a star with a small radius. Thus the waveform will have
greater asymmetries produced by Doppler effects if the star is large than if it is
small. The mass to radius ratio affects the fraction of the cycle when the spot is
visible; in the Newtonian limit R GM/c 2 exactly half the surface is visible,
but for radii appropriate to neutron stars light-bending effects make more of the
surface visible, and for a slowly rotating star with R < 3.5GM/c 2 the entire surface
can be seen. More compact stars will therefore tend to produce lower amplitude
waveforms. Hence in principle a waveform can be analyzed to infer the radius and
mass, as well as other quantities such as the rotational latitude of the spot center,
the rotational latitude of the observer, the spot angular radius (which turns out to be
unimportant, as does the spot shape, as long as the spot radius is significantly less
than the latitude of the spot center), and the emission pattern from the surface as
seen in the spot’s local rest frame.
Work by Muno et al. [160] initially appeared to cast doubt on this model of burst
oscillations, because when they stacked data from multiple bursts from a given star
they seemed to find that in some cases high energy photons arrive after low energy
photons, i.e., the hard photons lag the soft photons. This is unexpected in the rotating
hot spot model because as the spot rotates into view, the Doppler effect blueshifts
the spectrum and thus high energies should lead low energies. However, a careful
re-examination of the data for 4U 1636–536, which showed the strongest hard-lag
trend in [160], demonstrated that in fact the oscillation phase versus photon energy
is entirely consistent with the rotating hot spot model [12]. Statistical fluctuations,
and probably the consequences of stacking burst data, appear to have led to the
opposite conclusion in [160]. Thus currently it does seem that rotating hot spots are
consistent with the data on burst oscillations.
The most common assumption in such modeling is the “Schwarzschild plus
Doppler” approximation (e.g., [29, 154, 162, 235]), in which the star is assumed to
be spherical and the spacetime external to the star is assumed to be Schwarzschild
(nonrotating), but all the special relativistic effects associated with the rotation of
the surface are treated exactly. Full simulations that trace rays in the numerical
spacetimes appropriate for rotating objects have demonstrated that the waveforms
generated using the Schwarzschild + Doppler approximation are indistinguishable
from the waveforms in the full simulation when there are < ∼ 10 5 total counts and the
M. C. Miller
1.4.3 Modeling of Waveforms
Thermonuclear bursts that display brightness oscillations, and isolated millisecond
pulsars that are detectable in X-rays, have had their periodic waveforms analyzed
in attempts to constrain their masses and radii. The basic principle is simple: if one
makes the standard assumption that the oscillations are rotational modulations of the
flux produced by a hot spot fixed on the star, then the shape of the waveform encodes
information about the mass and radius (see [182], in which the relevant formulae
were derived and which was the first paper to use millisecond pulsar waveforms to
constrain the neutron star mass and radius). For example, a star of a given rotation
frequency that has a large radius will have a higher linear surface rotational speed
at a given latitude than a star with a small radius. Thus the waveform will have
greater asymmetries produced by Doppler effects if the star is large than if it is
small. The mass to radius ratio affects the fraction of the cycle when the spot is
visible; in the Newtonian limit R GM/c 2 exactly half the surface is visible,
but for radii appropriate to neutron stars light-bending effects make more of the
surface visible, and for a slowly rotating star with R < 3.5GM/c 2 the entire surface
can be seen. More compact stars will therefore tend to produce lower amplitude
waveforms. Hence in principle a waveform can be analyzed to infer the radius and
mass, as well as other quantities such as the rotational latitude of the spot center,
the rotational latitude of the observer, the spot angular radius (which turns out to be
unimportant, as does the spot shape, as long as the spot radius is significantly less
than the latitude of the spot center), and the emission pattern from the surface as
seen in the spot’s local rest frame.
Work by Muno et al. [160] initially appeared to cast doubt on this model of burst
oscillations, because when they stacked data from multiple bursts from a given star
they seemed to find that in some cases high energy photons arrive after low energy
photons, i.e., the hard photons lag the soft photons. This is unexpected in the rotating
hot spot model because as the spot rotates into view, the Doppler effect blueshifts
the spectrum and thus high energies should lead low energies. However, a careful
re-examination of the data for 4U 1636–536, which showed the strongest hard-lag
trend in [160], demonstrated that in fact the oscillation phase versus photon energy
is entirely consistent with the rotating hot spot model [12]. Statistical fluctuations,
and probably the consequences of stacking burst data, appear to have led to the
opposite conclusion in [160]. Thus currently it does seem that rotating hot spots are
consistent with the data on burst oscillations.
The most common assumption in such modeling is the “Schwarzschild plus
Doppler” approximation (e.g., [29, 154, 162, 235]), in which the star is assumed to
be spherical and the spacetime external to the star is assumed to be Schwarzschild
(nonrotating), but all the special relativistic effects associated with the rotation of
the surface are treated exactly. Full simulations that trace rays in the numerical
spacetimes appropriate for rotating objects have demonstrated that the waveforms
generated using the Schwarzschild + Doppler approximation are indistinguishable
from the waveforms in the full simulation when there are < ∼ 10 5 total counts and the