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accretion - powered millisecond pulsars ( amps ) provide a unique laboratory for studying the process of disk accretion onto magnetic stars ( @xcite and refrences therein ) . the @xmath0@xmath1  g stellar magnetic field truncates the accretion disk only a few neutron star radii above the surface , channeling the infalling matter toward the magnetic poles . the resulting x - ray  emitting hot spots and accretion shocks are modulated by the stellar rotation to produce the observed pulsations . such compact accretion geometry poses a particularly interesting challenge . we want to understand the details of the interaction between the magnetic field and the disk , and the effects that result from the proximity of different emitting regions . the accretion - powered 2.5  ms pulsar  is a particularly good system for this type of study : in addition to having multiple outbursts , it spans orders of magnitude in accretion rate . in this paper we report our study of the energy - dependence of pulsations from this source and show that it gives new insight into these problems . was the first detected amp @xcite , and it remains the best studied . persistent pulsations have been detected throughout the four @xmath01 month outbursts from this source that have been observed by the _ rossi x - ray timing explorer _ ( _ rxte _ ) during 19982005 . the timing of these pulsations establishes a 2.01  hr binary orbital period @xcite that is increasing on a timescale of @xmath2  yr , possibly due to the ablation of its low - mass companion during x - ray quiescence ( @xcite , hereafter h08 ; @xcite ) . the presence of x - ray pulsations during the peak accretion rate and the long - term spin - down of this source provide a tight constraint on its magnetic field : @xmath3@xmath4  g ( h08 ; @xcite ) . a distance of 3.43.6  kpc is estimated from thermonuclear x - ray burst properties @xcite .    the outbursts from  had similar light curves , which we divide for convenience into four stages ( see fig .  2 of h08  for a schematic ) : a rise from quiescence lasting @xmath55  days ; a short - lived peak at a 225  kev flux of @xmath6@xmath7  erg  @xmath8  s@xmath9 ; a 1015  day slow decay in luminosity that lasts until the source reaches @xmath5@xmath10  erg  @xmath8  s@xmath9 ; and finally a rapid fall in luminosity followed by weeks of low - luminosity flares with a @xmath05  d periodicity , during which the observed flux can vary by a factor of 1000 @xcite . this sudden drop has been interpreted as a transition into a `` propeller '' accretion regime ( e.g. , * ? ? ? * ) in which accretion is only partially inhibited @xcite , although it may also be caused by a change in the viscosity of the accretion disk as it transitions into quiescence @xcite .    shortly after the discovery of pulsations in , @xcite noted the presence of soft phase lags , the tendency of the pulses at lower ( `` softer '' ) energies to arrive later than the corresponding pulses at higher ( `` harder '' ) energies . the magnitude of the lags increased approximately linearly between 210  kev and saturated at 0.08 rotational cycles ( 200  @xmath11s ) above 10  kev . similar soft lags have been detected in six other amps : igr  j00291@xmath125934 @xcite , xte  j0929@xmath13314 @xcite , xte  j1751@xmath13305 @xcite , xte  j1807@xmath13294 @xcite , xte  j1814@xmath13338 @xcite , and hete  j1900.1@xmath132455 @xcite .    a number of explanations for the amp soft lags have been proposed . models that include only doppler boosting , due to the stellar rotation @xcite or the orbiting material in the accretion disk @xcite , have shortcomings that are summarized by @xcite . the two most plausible explanations invoke a two - component emission model with differing angular dependence and spectra ( @xcite ; hereafter pg03 ) or the scattering delay of hot photons from the accretion shock off the cooler disk or stellar surface ( @xcite ; hereafter ft07 ) . in this paper we present an energy - resolved analysis of the persistent pulsations for all the outbursts of  detected by _ rxte_. we will discuss the implications of our results for these models of the soft lags . for our study of the energy dependence of the persistent pulsations , we analyzed the same _ rxte _ observations of  as in h08  ( refer to table  1 therein ) . the ` e_125us_64m_0_1s ` data mode of the _ rxte_proportional counter array ( pca ; @xcite ) was used for nearly all the observations . it provides 122  @xmath11s  time bins and 64  energy channels divided evenly over the detector response , a good combination for discerning the energy dependence of the persistent pulsations . the energy band covered by each channel varied over the 7  yr of observation spanned by this study due to adjustments to the pca anode voltages @xcite ; we accounted for this effect throughout our measurements .    for our timing analysis , we shifted the photon arrival times to the solar system barycenter , applied the _ rxte _ fine clock correction , and filtered out data during earth occultations and intervals of unstable pointing . we also removed any data within 5  min of thermonuclear x - ray bursts . to measure the phases and fractional amplitudes of the persistent pulsations for a given energy band @xmath14 , we folded 512  s intervals of data using the phase timing solution derived in h08 . dividing the resulting pulse profiles into @xmath15 phase bins with photon counts @xmath16 , @xmath17 , the amplitude ( @xmath18 ) and pulse phase offset ( @xmath19 ) for the @xmath20th harmonicth harmonic is @xmath20 times the frequency of the 401  hz fundamental . ] is @xmath21 we measure the phase offsets of the energy bands relative to the softest band , @xmath22 : @xmath23 . thus negative values of @xmath24 indicate that pulsations of the @xmath20th harmonic within band @xmath14 arrived earlier than the pulsations within the softest band . the uncertainties of these phase offsets due to poisson noise are @xmath25 where @xmath26 is the total number of photons in the energy band ( h08 ) . for our analysis , we rejected phase measurements with @xmath27  cycles ( i.e. , 0.1  ms ) . the measured fractional rms amplitudes are @xmath28 where @xmath29 is the approximate number of background events within our energy band and time interval , estimated using the ftool pcabackest . uncertainties on the fractional amplitudes are computed using the method described by @xcite and @xcite , which correctly accounts for the addition of noise to the complex amplitude of the signal . we estimated the 2.525  kev fluxes of the persistent emission by fitting the spectra extracted from standard2 mode data ( 128 energy channels , 16  s time resolution ) with an absorbed blackbody plus power law model , including an iron line if there were significant residuals around 6.4  kev ; see @xcite ,  2.1 , for full details . ( relative to the 2.13.3 kev band ) and fractional amplitudes during the slow - decay stages of the 2002 outburst ( black ; mjd 52564.152575.0 ) and the 1998 outburst ( grey ; mjd 50919.050932.8 , the same pointed observations used by @xcite ) . negative offsets indicate soft lags . [ fig : avgpulsationsvsenergy],scaledwidth=45.0% ]        to assess the energy dependence of the pulse phase and fractional amplitudes , we calculated the average phase offsets and fractional amplitudes during the 11  d slow - decay stage of the 2002 outburst . figure  [ fig : avgpulsationsvsenergy ] shows the results , along with measurements from the 1998 outburst for comparison . we chose this time range because it was the longest and most data - rich interval during which the pulse profile of  was observed to be reasonably stable . the pulse profiles during the slow - decay stage of the 2005 outburst , although similar to the 2002 profiles , were somewhat less stable ( see fig .  3 of h08 ) . the soft phase lag of the pulsations is immediately apparent for the fundamental . between 2  kev and 10  kev , the pulses in harder bands arrive progressively sooner , with a slope of @xmath30  cycles  kev@xmath9 . between 10  kev and 20  kev , this trend saturates at a lead of 0.08  cycles ( 200  @xmath11s ) . there is impressive agreement between the 1998 and 2002 outbursts , which also had nearly identical light curves ; the phase lags during the dimmer 2005 outburst are similar in shape but plateau at a smaller magnitude . the phase offsets of the second harmonic are less pronounced , disappearing almost entirely or becoming hard lags when the source is bright , as is the case for the data in figure  [ fig : avgpulsationsvsenergy ] .    in contrast , the energy dependence of the fractional amplitudes varies considerably between outbursts . the fractional amplitude of the fundamental during the slow - decay stage of the 1998 outburst ( 5.27.5% rms ) was considerably higher than during the 2002 outburst ( 3.85.0% rms ) , as can be seen in figure  [ fig : avgpulsationsvsenergy ] ; the amplitude of the 2005 outburst ( 2.02.6% rms ) was lower still . the morphology of the energy dependence differs as well . the 1998 outburst exhibits a large fractional amplitude peak at 3.5  kev and falling amplitudes up to 20  kev , similar to what is seen in xte  j1814@xmath13338 @xcite . in contrast , during 2002 and 2005 the fractional amplitude vs. energy curves are much flatter , with small but significant peaks at 8  kev . given the strong similarities between the light curves of the 1998 and 2002 outbursts , it is curious that the energy dependence of their fractional amplitudes is markedly different . all three outbursts show weak evidence of the fundamental s fractional amplitude increasing again above @xmath520  kev . the fractional amplitude of the second harmonic generally increases with energy for all the outbursts , indicating less sinusoidal pulse profiles in harder energy bands . xmm observations of the 2008 outburst of this source reveal that the fractional amplitude of both harmonics continues to decrease below 2  kev @xcite . figure  [ fig : bigenergyplot ] illustrates how the energy - dependent phases and fractional amplitudes change over the course of the four observed outbursts . the soft lags in the fundamental are apparent throughout all four outbursts ; for the second harmonic , soft lags are apparent everywhere except during the outburst peaks . however , the degree of these lags is not constant : they are much more pronounced during the tails of the outbursts , when the flux is low . @xcite noted a similar pattern in xte  j1814@xmath13338 , although that source did not trace as wide a range of fluxes , so the relationship was less clear . the energy dependence of the fractional amplitudes does not exhibit any clear correlation with the source flux or any other measured parameters , apart from the general trend of fractional amplitudes increasing in the outburst tails ( see  3.4 of h08 ) .    to further understand the relationship between the soft lags and the flux , we measured the phase vs. energy slopes over 210  kev . we chose this range because it has the most significant energy dependence for the fundamental and , to a lesser extent , for the second harmonic . figure  [ fig : phaselagslopevsflux ] plots the resulting slopes as a function of flux . for the fundamental , the magnitude of the soft lags is clearly anticorrelated with the 225 kev flux when the source is bright : as the source dims from its peak at @xmath31  erg  @xmath8  s@xmath9  down to roughly @xmath10  erg  @xmath8  s@xmath9 , the magnitude of the soft lags grows . this range of fluxes covers the peak and slow - decay stages of the outburst . at lower fluxes , the relation reverses , and the magnitude of the lags becomes correlated with flux . the flux at which this turnover occurs is significant : once it falls to @xmath10  erg  @xmath8  s@xmath9 , the source dims rapidly and enters the flaring - tail stage of the outbursts . erg  @xmath8  s@xmath9  turnover . these fits are based on the 2002 and 2005 data only , as the 1998 and 2000 outbursts exhibit greater scatter . [ fig : phaselagslopevsflux],scaledwidth=40.0% ]    the trend for the second harmonic is somewhat less clear . measuring a slope through the 210  kev fractional amplitude points does not work as well since the energy dependence is less linear ( top panel of fig . [ fig : avgpulsationsvsenergy ] ) , so there is more scatter . even so , the data strongly suggest a similar relationship at high fluxes , particularly when the relatively scattered 1998 data are excluded . the slopes of these trends differ significantly for the two harmonics . in the low - flux regime , the slopes are @xmath32 for the fundamental and @xmath33 for the second harmonic ; at high fluxes , they are @xmath34 for the fundamental and @xmath35 for the harmonic . ( the units in all cases are @xmath36  cycles  kev@xmath9  [ log  flux]@xmath9 . ) these slopes were fit using the 2002 and 2005 data only , as the two earlier outbursts had appreciably more scatter and less data . note that these units reflect _ rotational _ cycles of the star , so a mechanism that induces the same phase lag in all photons of a given energy would result in the slopes being equal for the two harmonics . for the high - flux slopes , we can conclude with 8@xmath37 significance that this is not the case . we also looked for correlations between the lags and the spectral parameters from our blackbody plus power law fits . the only parameter for which a correlation is apparent is the normalization of the hard power law component . this relation is not surprising , because the hard normalization and the total x - ray flux are linearly proportional : during the 19982005 outbursts , @xmath38 , with a reduced @xmath39 near unity . the uncertainties of @xmath40 are much greater than the flux uncertainties , particularly when the source is dim , and the large scatter obscures whether a turnover is present in the lag vs. @xmath40 relation . the energy dependence of the pulsations from  exhibit strong variability both between and during the four outbursts observed by _ rxte_. the overall fractional amplitudes and the shape of the amplitude vs. energy curves differ substantially between outbursts , with no obvious correlation with flux ; the only clear trend is that the pulsations generally become less sinusoidal with energy in the observed 230  kev range . in contrast , the soft lags of the pulses have a consistent energy dependence , increasing in magnitude in the 210  kev band and saturating at 1030  kev , with a strong dependence on flux . at the highest fluxes the lags are near zero , but they increase steadily as the flux falls . below a 225  kev flux of @xmath5@xmath41  erg  @xmath8  s@xmath9 , this trend reverses and lags get smaller again . this particular flux also marks the point in the outburst where the luminosity decays rapidly and the source enters the flaring tail . soft lags are a common feature of amp pulsations , and two plausible models have emerged to explain their existence . pg03 suggested a two - component model involving soft emission from a surface hot spot and a hard comptonized component from the shocked layer in the accretion funnel . each component has a different beaming pattern , which is affected in a different way by the doppler effect . this results in the soft pulse lagging the hard pulse . while this model works well for  and xte  j1751@xmath13305 @xcite , it does not predict the reduction in soft lags at very high energies seen in igr  j00291@xmath125934 @xcite . as a result ft07 proposed an alternative model in which the disk plays a prominent role . in this model , the main pulsed element is the hard component from the accretion column : this radiation is then downscattered by cold plasma in the inner disk ( or on the neutron star surface ) to generate a soft lagging component.300  ev in the inner disk @xcite ; the actual temperature will be higher due to the irradiation . at these temperatures and the expected electron densities ( @xmath42  @xmath43 ; see ft07 and fig .  [ fig : falangafitvsmdot ] here ) , all elements through oxygen will easily be fully ionized by the hard tail of the thermal distribution . the only significant source of photoelectric absorption is fe , but for this @xmath44 only fe  xxv and higher remain at @xmath45  ev , with the transition to full ionization occuring at 400600  ev @xcite . its he - like and h - like forms have k edges at 9  kev , so they will not absorb the softer photons responsible for the lags . since the ft07 model is not ruled out _ a priori _ , this discussion attempts to address both models without preference . ] these soft lags get larger as the density of the scattering plasma falls . let us now explore the factors that might cause the flux dependence of the soft lags , assuming that flux directly traces accretion rate . we then need to ask how the properties of the system are expected to change as the accretion rate falls . for a star with a magnetic field that is strong enough to channel the accretion flow , the reduction in accretion rate should cause an increase in inner disk radius ( e.g. , @xcite ) . this will have two relevant effects . first , the density of scattering plasma near the star will fall : within the model of ft07 this would cause the soft lags to increase . in addition the relative angular velocity of the inner disk and star will reduce . this will affect the angle at which the accretion funnel impacts the surface @xcite , changing the column density above any surface hot spot . this type of change might work within the model of pg03 to affect the soft lags .    if these changes continue steadily , however , it is hard to see what could cause the turnover seen in figure  [ fig : phaselagslopevsflux ] . one intriguing possibility that would give a very natural explanation for such a turnover is that the system reaches the point where the magnetospheric radius @xmath46 exceeds the corotation radius @xmath47 . this is often referred to as the `` propeller '' regime , since it was originally thought that matter would then be ejected from the system , with the star going from spin - up to spin - down @xcite . in fact , the transition is probably not sharp , and @xmath46 needs to be substantially larger than @xmath47 for ejection to occur @xcite . at intermediate accretion rates , continued accretion is possible even while the neutron star starts to spin down @xcite . simulations by @xcite indicate that the transition between spin up and spin down should occur when @xmath48 for amps . the observed turnover happens at an accretion rate of @xmath49  yr@xmath9 , giving @xmath50 for the magnetic field @xmath51  g from h08 . , using @xmath52  kpc and the bolometric correction @xmath53 from @xcite . the alfvn radius was estimated as @xmath54 , with @xmath55 as per @xcite . @xmath56 and @xmath57  km were used in all cases . ] this close agreement between the simulations and the @xmath58 at which the flux rapidly drops and the soft lags turnover strongly suggests that  passes through this transition as the accretion rate falls . in this eventuality there are two effects that could cause the turnover in soft lags .     of the inner accretion disk , as fit using the model of ft07 , versus the mass accretion rate for the 2002 and 2005 outbursts of . the vertical dotted line marks the @xmath58 at which the magnitude of the soft lags is greatest and below which the outburst is in its flaring - tail stage ; a transition from normal accretion to the `` propeller '' regime can explain this change in the observed behavior . the dashed lines give the best fits of @xmath59 in each region . [ fig : falangafitvsmdot],scaledwidth=47.0% ]    the first is due to the fact that the relative angular velocity of the inner disk and the star changes sign . the accretion funnel will go from being dragged forwards to being dragged backwards ( assuming that it settles rapidly ; @xcite ) . the column depth above any surface hot spot should therefore go through a maximum near the zero torque point , causing a turnover in the scattering properties of the accretion column . if the accretion column plays a major role as discussed in pg03 , this could cause the observed turnover . another explanation within the pg03 model is the suppression of the lagged soft component at high fluxes due to the increased optical depth of the accretion shock when @xmath58 is large . however , this mechanism would require the normalization of the soft component to decrease during the outburst peaks , contrary to what is seen ( cf . fig .  2 of @xcite ) . a second possibility stems from the fact that the properties of the disk must also change at this point . for accretion to continue after zero - torque point , the inner edge stops retreating , and the density at the inner edge must rise @xcite . in the scenario of ft07 , this rise in inner disk density would cause the magnitude of the soft lags to decrease again . in the absence of compton upscattering , which would introduce hard lags not observed in , their model predicts that the scattering from the disk will modify the arrival times as @xmath60 , where @xmath61 is the thompson cross - section and @xmath44 is the electron density of the scattering region . this model provides reasonable fits to the observed phase lags ; figure  [ fig : falangafitvsmdot ] shows the resulting @xmath44 .    the fitted electron densities probe the relationship between the inner disk and the accretion rate . @xmath62 @xmath43 gives a mean free path of @xmath63  km and time @xmath64  @xmath11s , requiring @xmath0100 scattering events to produce the observed soft lags . this @xmath65 rules out the thin neutron star atmosphere as a scattering medium . on the other hand , simulations by @xcite suggest an inner disk thickness of order the neutron star radius , and @xmath66 is more than adequate to accomodate the number of needed scattering events . from the fits in figure  [ fig : falangafitvsmdot ] , we find that @xmath67 during the main outburst ; during the presumed propeller stage , @xmath68 . an interesting consequence of the former relationship is that at higher accretion rates before the turnover , when @xmath69 , the flow rate ( the volume of material per unit time crossing the inner edge of the disk ) must be constant : since @xmath70 , where @xmath71 is the solid angle of the inner edge and @xmath72 is the radial velocity of material crossing it , the flow rate @xmath73 is fixed . if the phase lags in  are entirely due to compton downscattering , as suggested by ft07 , then this model must explain why the magnitudes of the soft lags for the second harmonic are roughly half the magnitudes for the fundamental over a wide range of @xmath58 , as can be seen in figure  [ fig : phaselagslopevsflux ] . following the formalism of @xcite , we model the timing effects of comptonization with a distribution @xmath74 of the photon escape times from the scattering medium . for the @xmath20th harmonic of the spin frequency @xmath75 , we convolve the harmonic component @xmath76 with @xmath74 to find the new amplitude @xmath77 and the time lag @xmath78 of the scattered pulses : @xmath79 in general , the choice of @xmath74 accounts for the geometry of the emitting and scattering regions . however , for the large number ( @xmath0100 ) of scatterings previously noted , the geometry becomes unimportant and a photon diffusion distribution of @xmath80 is appropriate @xcite . applying this form to equation  ( [ eq : st80formalism ] ) , the time lag of the @xmath81 harmonic is @xmath82 the diffusion timescale can not be longer than the lags themselves : @xmath83  @xmath11s . this sets a lower limit of @xmath84 due to scattering alone , contrary to the observation that @xmath85 . thus compton scattering can not by itself fully explain the observed lags : a change in the energy dependence of the initial pulses is still necessary . clearly , detailed calculations are required to check whether the two models really would predict the outcomes that we suggest , but they are at least in principle plausible . one would also need to test whether the models would predict any reversal in other pulsation properties at propeller onset , given that no such changes are observed . the interesting behavior of the soft lags in this source suggests that the lags of the other accretion - powered millisecond pulsars should be revisited to check for flux dependence . the only source for which this has been done is xte  j1814@xmath13338 : an increase in soft lags was observed in the tail of the outburst @xcite , but the rapid drop off and shorter tail of this source made it very difficult to study the pulsations at the lowest fluxes . the other amps may be more promising as candidates for further study . jmh thanks the max planck institut fr astrophysik for hospitality that facilitated the pursuit of this research . we would also like to thank lev titarchuk , juri poutanen , and j. martin laming for useful conversations . this work was supported at mit in part by nasa grants under the rxte guest observer program .                                              , k. , swank , j.  h. , giles , a.  b. , stark , m.  j. , strohmayer , t. , zhang , w. , & morgan , e.  h. 1996 , in proc . 2808 , p. 59 - 70 , euv , x - ray , and gamma - ray instrumentation for astronomy vii , oswald h. siegmund ; mark a. gummin ; eds . , o.  h. siegmund & m.  a. gummin , 5970
soft phase lags , in which x - ray pulses in lower energy bands arrive later than pulses in higher energy bands , have been observed in nearly all accretion - powered millisecond pulsars , but their origin remains an open question . in a study of the 2.5  ms accretion - powered pulsar , we report that the magnitude of these lags is strongly dependent on the accretion rate . during the brightest stage of the outbursts from this source , the lags increase in magnitude as the accretion rate drops ; when the outbursts enter their dimmer flaring - tail stage , the relationship reverses . 
 we evaluate this complex dependence in the context of two theoretical models for the lags , one relying on the scattering of photons by the accretion disk and the other invoking a two - component model for the photon emission . in both cases 
 , the turnover suggests that we are observing the source transitioning into the `` propeller '' accretion regime .