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strong x - ray pulsations with a 2.49 msec period were discovered from sax j1808.4 - 3658 + = in an april 1998 observation with the rossi x - ray timing explorer @xcite . the pulsar is a member of an accreting binary system with an orbital period of 2.01 hour and a low - mass companion @xcite . though similar to other low - mass x - ray binaries in its timing and spectral properties ( e.g. * ? ? ?
* ; * ? ? ?
* ) , sax j1808.4 - 3658 + = is unique for its x - ray pulsations .
no other such binary has shown coherent pulsations in its persistent flux despite of careful searches ( * ? ? ? * and references therein ) .
as such , sax j1808.4 - 3658 + = is the fastest rotating accreting neutron star .
if the pulsations are due to modulated emission from one hot spot on the neutron star surface , the 2.49 msec period corresponds to an equatorial speed of approximately 0.1c . with these high speeds , sax j1808.4 - 3658 +
= offers an excellent system for studying relativistic effects .
one such effect may be the observed lag of low - energy photons relative to high - energy photons in the pulse discovered by @xcite .
@xcite suggest that the lags are due to comptonization in a relatively cool surrounding medium .
alternatively , the lags may be the result of a relativistic effect : the high - energy photons are preferentially emitted at earlier phases due to doppler boosting along the line of sight .
this possibility was suggested for the similar lags in the 549  hz oscillations in an x - ray burst of aql  x-1 @xcite , where a simple model showed that the delays roughly match those expected .    in the following , we present new measurements of the pulsed emission from sax j1808.4 - 3658 + = .
we show that the energy - dependent phase lags are equivalent to a hardening pulse profile .
we model this behavior in terms of a hot spot on the neutron star , including relativistic effects .
we have used publicly available data from the proportional counter array ( pca ) on board rxte in an ` event ' mode with high time resolution ( 122@xmath2sec ) and high energy resolution ( 64 channels ) .
the observations occurred from april 10 1998 to may 7 1998 , when the source was in outburst .
we generate folded lightcurves in each pca channel .
this is accomplished with the fasebin tool in ftools version 4.2 , which applies all known xte clock corrections and corrects photon arrival times to the solar system barycenter using the jpl de-200 ephemeris , yielding a timing accuracy of several @xmath2sec ( much less than the phase binning used here ) . as a check ,
we have applied this method to crab pulsar data and the results are identical to @xcite . to produce pulse profiles in the neutron star rest frame , we use the sax j1808.4 - 3658 + = orbital ephemeris found by @xcite .
an example folded lightcurve is shown in figure  [ lc ] ( top ) for the observation of april 18 1998 14:05:40 to april 19 1998 00:51:44 utc .
to study the energy spectra at each phase bin , we take the rates at pulse minimum and subtract it from the rest of the data at other phases .
this effectively accomplishes background subtraction and eliminates the unpulsed emission which we do not wish to consider .
note , however , that the pulsed emission may have some contribution even at the pulse minimum and this subtraction scheme represents only a best approximation to the true pulse emission .
we generate detector response matrices appropriate to the observation date and data mode using pcarsp v2.38 , and use xspec v.10.0 to fit model spectra .
we fit the spectrum here with a simple powerlaw function .
though the function itself is not meant to be a physical description , the powerlaw index provides a good measure of the spectral hardness .
fits in each phase bin have reduced @xmath3 of 0.7 to 1.6 . including an interstellar absorption does not substantially affect these results .
the powerlaw index clearly increases through the pulse phase ( figure  [ lc ] , bottom ) , i.e. spectrum evolves from hard to soft .
we also fit the profiles of the folded lightcurves in each channel using fourier functions at the fundamental frequency and its harmonics . from these fits
we determine the phase lag in each channel relative to the fits in some baseline channel range .
results for the 18 april 1998 observation are shown in figure  [ lags ] as solid symbols .
note that negative numbers indicate that high - energy photons precede low - energy photons .
we are also able to measure lags in the first harmonic , and find that they are opposite in sign to the fundamental , i.e. low - energy photons precede high - energy photons in the first harmonic .
no lags are measurable in the other harmonics .
another way to measure energy - dependent phase lags is by fourier cross - correlation analysis .
this is the method used for sax j1808.4 - 3658 + = by @xcite and for other timing signals as well , e.g. kilohertz qpos ( e.g. * ? ? ?
for a description of cross correlation analysis see e.g. @xcite . from the pca event mode data we calculate fourier spectra in various channel ranges with nyquist frequency of 2048 hz and resolution of 0.25 hz .
we then calculate cross spectra defined as @xmath4 , where @xmath5 are the complex fourier coefficients for two energy bands at the pulsar frequency @xmath6 .
the phase lag between the two energy bands is given by the argument of @xmath7 .
we measure all phase delays relative to the unbinned channels range 5 to 8 , i.e. 1.83 to 3.27 kev for 5 detector units in pca gain epoch 3 ( april 15 1996 to march 22 1999 ) .
the results for the 18 april 1998 observation , are shown in figure  [ lags ] by the open symbols .
the phase lags are consistent with those calculated from the lightcurve fitting . from the cross - correlation spectra
we are not able to measure lags in the much weaker harmonics .
we have calculated phase lag spectra also for other rxte observations during the april 1998 outburst .
these spectra are similar to that in figure  [ lags ] .
to quantify the trends , we compute an average phase delay , @xmath8 , over all energies for each observation .
we also fit a broken powerlaw function to each phase delay spectrum : @xmath9 below a break energy , @xmath10 , and @xmath11 above this energy .
figure  [ lags_v_time ] shows the quantities @xmath8 , @xmath10 and @xmath12 versus the time of each observation .    there is a clear connection between the results of the two analyses presented here .
the phase resolved spectroscopy shows that the spectrum softens and correspondingly the peak of the pulse profile appears slightly earlier in phase for higher energies ( figure  [ lc ] ) .
the method of measuring phase delays shows the same behavior : higher - energy photons preferentially lead lower - energy photons in the fundamental and the magnitude of this phase delay increases with energy ( figure  [ lags ] ) . in the following we discuss a model that can account for the phase delays / spectral softening measured here .
we calculate the expected luminosity as a function of phase in a manner similar to @xcite but including doppler effects @xcite and time of flight delays @xcite .
this treatment is based on a schwarzschild metric , where the photon trajectories are completely determined by the compactness , @xmath13 .
the predicted luminosity as a function of phase from our code matches the results of @xcite and @xcite for the various choices of parameters .
@xcite recently developed a model for pulse profiles using a slightly different approach .
the parameters in the model are the speed at the equator of the neutron star , @xmath14 , the mass of the neutron star , @xmath15 , the compactness , @xmath13 , the angular size of the cap , @xmath16 , and the viewing angles , @xmath17 ( the angle between the rotation axis and the cap center ) and @xmath18 ( rotation axis to line of sight ) .
another ingredient is the emission from the spot , which we take as isotropic and isothermal .
the spectrum of energy emitted from the spot is another important input .
note that if the emitted spectrum is power law like , the observed spectrum will not evolve with phase , since doppler transformation preserves the shape ( see * ? ? ?
the intrinsic spectrum must therefore have some shape that transforms to match the observed hardening and phase lags ; we use a blackbody spectrum with temperature @xmath19 .
a fit from the model is shown in figure  [ lags ] .
this fit uses the following model parameters : @xmath20 , @xmath21\c\else\if'\c\else\space\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}\else\if\empty\c\else\if\space\c\else\space\fi\fi\fi}\fi}}}$ ] , @xmath22 kev , @xmath23 , @xmath24 , @xmath25 . to derive count rates , we use table models in xspec and the appropriate response files as discussed above .
the fit for this single set of parameters is good ; we find @xmath26 from the fourier cross - correlation data or a reduced @xmath3 of 0.8 for all the parameters fixed .
a full exploration of the parameter space of the model is beyond the scope of this letter .
we note the following trends , however .
the magnitude of the phase lags depends sensitively on @xmath14 and @xmath19 .
larger delays result from higher speeds , because the pulses become increasingly asymmetric due to doppler boosting .
this asymmetry is also energy dependent , so there is a dependence on @xmath19 as well ( see * ? ? ?
the lags also depend on @xmath17 and @xmath18 , especially at higher energies where there is the turn - over noted above .
the phase lags are less sensitive to @xmath13 and @xmath16 .
we have tested the assumption of isotropic emission and found that the phase lags depend only weakly on the beaming .
the pulsed emission from sax j1808.4 - 3658 + = evolves through its phase from a relatively hard to a soft spectrum , as shown by our phase resolved spectroscopy ( figure  [ lc ] ) .
this evolution can also be thought of , and measured as , an energy - dependent phase lag in the fundamental ( figure  [ lags ] ) , i.e. higher - energy photons emerging earlier in phase than lower - energy photons .
we have applied a model to the data which consists of a hot spot on the rotating neutron star under a general relativistic treatment .
the dominant effect accounting for energy - dependent delays is doppler boosting , the larger boosting factors at earlier phases giving harder spectra .
the model fits the data quite well .
the model also provides a stable mechanism for generating the phase delays , which meshes with the fact that the characteristic delays remain stable in time to within 25% ( figure  [ lags_v_time ] ) even as there is a factor of two decrease in the x - ray flux , a possible tracer of accretion rate . as noted in @xcite , in addition to explaining the energy - dependent phase lags in sax j1808.4 - 3658 + = , this mechanism may also account for the lags in the burst oscillations of aql x-1 @xcite and kilohertz qpos @xcite .
phase resolved spectroscopy has not yet been possible in these signals .
the model offers a new means of measuring the neutron star mass and radius , a notoriously difficult problem in accreting binary systems . the radius is directly related to @xmath14 , the equatorial speed ( @xmath27 ) .
the fits also depend on the model parameters @xmath15 and @xmath13 , though the lag spectra are less sensitive to these quantities .
the model could be independently constrained by future optical spectroscopy of the companion which could provide information on @xmath15 and the viewing angles , @xmath16 and @xmath17 .
we acknowledge stimulating discussions with the participants of the august 1999 aspen workshop on relativity where an early version of this work was presented .
we thank michiel van der klis , mariano mndez and luigi stella for helpful discussions .
this work was supported by nwo spinoza grant 08 - 0 to e.p.j.van den heuvel , by the netherlands organization for scientific research ( nwo ) under contract number 614 - 51 - 002 , and by the netherlands researchschool for astronomy ( nova ) .
this research has made use of data obtained through the high energy astrophysics science archive research center online service , provided by the nasa / goddard space flight center .
we analyze the properties of the pulsed emission from the accreting millisecond pulsar sax j1808.4 - 3658 + = in observations of its april 1998 outburst by the rossi x - ray timing explorer . 
 pulse phase spectroscopy shows that the emission evolves from a hard spectrum ( power law with photon index @xmath0 ) to a soft spectrum ( index @xmath1 ) . 
 this softening is also observable as a phase lag in the fundamental of low - energy photons with respect to high - energy photons . 
 we show that this lag is roughly constant over ten days of the outburst . 
 we fit these data with a model where the pulse emission is from a hot spot on the rotating neutron star and the flux as a function of phase is calculated including the effects of general relativity . 
 the energy - dependent lags are very well described by this model . 
 the harder spectra at earlier phases ( as the spot approaches ) are the result of larger doppler boosting factors which are important for this fast pulsar .