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Paper text:
shortly after the launch of the _ rossi _ x - ray timing explorer ( rxte ) in late 1995 , single kilohertz brightness oscillations were discovered in rxte countrate time series data from thermonuclear x - ray bursts in several neutron - star low - mass x - ray binaries .
these oscillations are remarkably coherent and their frequencies are very stable from burst to burst in a given source @xcite .
these oscillations are therefore thought to be at the stellar spin frequency or its first overtone .
this suggests that the oscillations are caused by rotational modulation of a hot spot produced by non - uniform nuclear burning and propagation .
analysis of these oscillations can therefore constrain the mass and radius of the star and yield valuable information about the speed and type of thermonuclear propagation . in turn , this has implications for strong gravity and dense matter , and for astrophysical thermonuclear propagation in other contexts , such as classical novae and type ia supernovae .    a comparison of theoretical waveforms with the observations
is required to extract this fundamental information . here
we exhibit waveform calculations that we have produced using general relativistic ray - tracing codes .
we survey the effects of parameters such as the spot size , the stellar compactness , and the stellar rotational velocity , and demonstrate that our computations fit well the phase lag data from sax  j18083658 .
to compute observed light curves , we do general relativistic ray tracing from points on the surface to the observer at infinity in a way similar to , but more general than , @xcite and @xcite . for simplicity , we assume that the exterior spacetime is schwarzschild , that the surface is dark except for the hot spot or spots , and that there is no background emission
. the amplitudes would be reduced by a constant factor if there were background emission .
the angular dependence of the specific intensity at the surface depends on both radiation transfer effects and doppler boosting ( see @xcite ) . for each phase of rotation
we compute the projected area of many small elements of a given finite size spot .
we then build up the light curve of the entire spot by superposing the light curve of all the small elements .
the grid resolution of the spot is chosen so that the effect of having a finite number of small elements can alter the value of the computed oscillation amplitudes by a fraction no larger than @xmath0 .
after computing the oscillation waveform using the above approach , we fourier - analyze the resulting light curve to determine the oscillation amplitudes and phases as a function of photon energy at different harmonics .
0.2truein          panel ( a ) of figure  1 shows the fractional rms amplitudes at the first two harmonics as a function of spot size and stellar compactness for one emitting spot centered on the rotational equator , as seen by a distant observer in the rotational plane .
these curves demonstrate that a common result of the hot - spot model is large - amplitude brightness oscillations with a high contrast in strength between the dominant harmonic and weaker harmonics , as is observed in several sources . the curves for the first harmonic illustrate the general shape of most of the first harmonic curves .
initially , the amplitude depends only weakly on spot size .
however , once the spot grows to an angular radius of @xmath1 there is a steep decline in the oscillation amplitude which flattens out only near the tail of the expansion .
this expected behavior appears to be in conflict with the decline in amplitude observed by strohmayer , zhang , & swank ( 1997 ) from 4u  172834 , in which the initial decline is steep .
the cause of this could be that the initial velocity of propagation is large , or that the observed amplitude is diminished significantly by isotropization of the beam due to scattering ( weinberg , miller , & lamb 1999 ) .
panel ( b ) of figure  1 shows the fractional rms amplitude at the second harmonic under the same assumptions but for two identical , antipodal emitting spots .
the range in spot size here is @xmath2 since two antipodal spots of @xmath3 radii cover the entire stellar surface .
note that in this situation , there is no first harmonic .
these figures show that when there is only one emitting spot , the fundamental is always much stronger than higher harmonics .
thus , a source such as 4u  1636536 with a stronger first overtone than fundamental @xcite must have two nearly antipodal emitting spots .
as described in detail in @xcite , we confirm the results of @xcite and @xcite that the rms amplitude decreases with increasing compactness until @xmath4 , then increases due to the formation of caustics .
we also find that a finite surface rotational velocity increases the amplitude at the second harmonic substantially , while having a relatively small effect on the first harmonic ( left panel of figure  2 ) .    as an application to data , in the right panel of figure  2
we use our models to fit phase lag data from the millisecond accreting x - ray pulsar sax  j18083658 .
the waveforms from the accreting spot are expected to be similar to the waveforms from burst brightness oscillations , and the signal to noise for this source greatly exceeds that from burst sources such as aql  x-1 @xcite . as is apparent from the figure
, the fit is excellent .
further data , especially from a high - area timing mission , could be used to constrain the stellar mass or radius from phase lag data .
many thermonuclear x - ray bursts exhibit brightness oscillations . 
 the brightness oscillations are thought to be due to the combined effects of non - uniform nuclear burning and rotation of the neutron star . 
 the waveforms of the oscillations contain information about the size and number of burning regions . 
 they also contain substantial information about the mass and radius of the star , and hence about strong gravity and the equation of state of matter at supranuclear densities . 
 we have written general relativistic ray - tracing codes that compute the waveforms and spectra of rotating hot spots as a function of photon energy . using these codes , we survey the effect on the oscillation waveform and amplitude of parameters such as the compactness of the star , the spot size , the surface rotation velocity , and whether there are one or two spots . 
 we also fit phase lag versus photon energy curves to data from the millisecond x - ray pulsar , sax  j18083658 .