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A Characterization of the Brightness Oscillations During Thermonuclear Bursts From 4U 1636-

a r X i v :a s t r o -p h /9904093v 1 8 A p r 1999A Characterization of the Brightness Oscillations During Thermonuclear

Bursts From 4U 1636–536

M.Coleman Miller

Department of Astronomy and Astrophysics,University of Chicago

5640South Ellis Avenue,Chicago,IL 60637,USA

miller@https://www.sodocs.net/doc/994731576.html,

ABSTRACT The discovery of nearly coherent brightness oscillations during thermonuclear X-ray bursts from six neutron-star low-mass X-ray binaries has opened up a new way to study the propagation of thermonuclear burning,and may ultimately lead to greater understanding of thermonuclear propagation in other astrophysical contexts,such as in Type Ia supernovae.Here we report detailed analyses of the ~580Hz brightness oscillations during bursts from 4U 1636–536.We investigate the bursts as a whole and,in more detail,the initial portions of the bursts.We analyze the ~580Hz oscillations in the initial 0.75seconds of the ?ve bursts that were used in a previous search for a brightness oscillation at the expected ~290Hz spin frequency,and ?nd that if the same frequency model describes all ?ve bursts there is insu?cient data to require more than a constant frequency or,possibly,a frequency plus a frequency derivative.Therefore,although it is appropriate to use an arbitrarily complicated model of the ~580Hz oscillations to generate a candidate waveform for the ~290Hz oscillations,models with more than two parameters are not required by the data.For the bursts as a whole we show that the characteristics of the brightness oscillations vary greatly from burst to burst.We ?nd,however,that in at least one of the bursts,and possibly in three of the four that have strong brightness oscillations throughout the burst,the oscillation

frequency reaches a maximum several seconds into the burst and then decreases.This behavior has not been reported previously for burst brightness oscillations,and it poses a challenge to the standard burning layer expansion explanation for the frequency

changes.

Subject headings:X-rays:bursts —stars:neutron

1.INTRODUCTION

Shortly after the launch of the Rossi X-ray Timing Explorer (RXTE )in December 1995,observation with RXTE of neutron-star low-mass X-ray binaries (LMXBs)revealed that several sources had a single,highly coherent,high-amplitude brightness oscillation during at least one

thermonuclear X-ray burst(for reviews see Strohmayer,Zhang,&Swank1997;Strohmayer, Swank,&Zhang1998a).The asymptotic frequency of these oscillations in the tails of bursts is so similar in di?erent bursts from a single source,and the oscillation is so coherent in the tail(see, e.g.,Strohmayer&Markwardt1999),that it is almost certain that this asymptotic frequency is the stellar spin frequency or its?rst overtone.These burst oscillations therefore provided the

?rst direct evidence for the value of the spin frequencies of these LMXBs,and they corroborate strongly the proposed evolutionary link between LMXBs and millisecond rotation-powered pulsars. In addition,the stability of the frequency in the tails of the bursts has led to their application as promising probes of the binary systems themselves(Strohmayer et al.1998b).

The existence of burst oscillations indicates that the emission from the surface,and hence the thermonuclear burning,is not uniform over the entire star.This is in accord with theoretical expectations(Joss1978;Ruderman1981;Shara1982;Livio&Bath1982;Fryxell&Woosley 1982;Nozakura,Ikeuchi,&Fujimoto1984;Bildsten1995),and it suggests that the properties of the burst oscillations,such as the evolution of their frequency or amplitude,may contain valuable information about the propagation of thermonuclear burning over the surface of the neutron star.The lessons learned from study of the thermonuclear propagation in bursts may ultimately further our understanding of thermonuclear propagation in other astrophysical contexts,such as classical novae and Type Ia supernovae.Unlike in novae or Type Ia supernovae,burning in thermonuclear X-ray bursts occurs near the surface and occurs often for a single source,and is therefore relatively easy to observe.The detailed study of burst brightness oscillations therefore has broad importance.

Here we describe in detail the frequency behavior of the burst oscillations in?ve bursts from 4U1636–536,which is an LMXB with an orbital period of3.8hours(see,e.g.,van Paradijs et al.1990).This source is of special interest because it produces detectable signals at both the fundamental and the?rst overtone of the stellar spin frequency(Miller1999),and because near the beginning of one burst the brightness oscillations reached the highest amplitude—50%rms—so far recorded for oscillations during a thermonuclear burst(Strohmayer et al.1998c).

In§2we analyze the light curves of the bursts,and the frequency and amplitude of the brightness oscillations in the four of those?ve bursts that have strong brightness oscillations for most of the duration of the burst.We?nd that,despite apparent similarities in the light curves of three of those four bursts,the amplitude and frequency behavior of their brightness oscillations are very di?erent from each other.We also?nd compelling evidence in one burst,and strong evidence in another burst,for an interval in which the burst oscillation frequency decreases after the peak in the light curve.

In§3we focus on the initial portions of the bursts.Analyses of bursts from many sources have shown that the oscillation frequency often changes by a few Hertz over the?rst few seconds of a burst(see Strohmayer et al.1998a for a review).The change is often a monotonic rise,but there are indications of more complicated behavior in some bursts.It has been pointed out that the

Table1:Starting Time of Bursts

Burst Date and Time

magnitude of the frequency change could be explained by a20–50meter expansion of the burning layers followed by a slow settling,if the layers conserve angular momentum(see,e.g.,Strohmayer et al.1998a),but details have not been worked out.For example,it is not clear how the layers would maintain their coherence throughout the5–10complete circuits relative to the body of the star that are implied by the observations.Bildsten(1998)has suggested that the layers may be stabilized by thermal buoyancy or mean molecular weight strati?cation,but details have not been worked out.In§3we examine in detail the?rst0.75seconds of all?ve bursts,which was the interval used to construct the candidate waveform for the~290Hz oscillation in4U1636–536 (Miller1999).We examine models of the frequency behavior that have increasing complexity:a constant-frequency model;one with a frequency and frequency derivative;a four-parameter model with an initial frequency and frequency derivative followed by a di?erent frequency derivative after a break time;and a?ve-parameter model with two di?erent frequencies and frequency derivatives separated by a break time.We?nd that if the same type of frequency model applies to all?ve of the bursts then the data do not require a model more complicated than the constant-frequency model or,possibly,the model with a single frequency and frequency derivative.Note,however, that this is not inconsistent with the use of the?ve-parameter model to construct a waveform used in the search for the expected~290Hz oscillation(Miller1999);in such a search,the only goal is to?nd the best?t to the~580Hz oscillations,and the extra parameters need not be justi?ed by a signi?cantly better?t.

Finally,in§4we discuss the implications of these results for the current picture of the frequency changes,in which the frequency change occurs because the burning layer is lifted by 20–50meters from the surface by the radiation?ux.We?nd that the simplest version of this picture has di?culty explaining the observations.

2.OVER VIEW OF THE BURSTS

We used public-domain data from the High Energy Astrophysics Science Archive Research Center.The data were taken in Single Bit Mode,which does not record the energy of photons. We give the starting times of the bursts in Table1and the light curves in Figure1.In burst d, the data dropouts are caused by telemetry saturation.

Fig.1.—Light curves for the bursts.(a)Burst beginning at22:39:24UTC on28December1996.

(b)Burst beginning at23:54:02UTC on28December1996.(c)Burst beginning at23:26:46UTC on29December1996.(d)Burst beginning at17:36:52UTC on31December1996.(e)Burst beginning at09:57:26UTC on23February1998.The data gaps in burst d are caused by telemetry

saturation.

Figure2shows the peaks of the power spectra of the?rst four bursts,as a function of time.The burst on23February1997does not have a strong brightness oscillation for most of its duration,and we therefore do not analyze it in the rest of this section.For each burst,the frequency of maximum power in successive nonoverlapping one-second intervals is shown by the solid triangles,and the Leahy et al.(1983)-normalized power at this frequency is shown by the solid line.Here we plot only those points with Leahy powers in excess of10(chance probability for a single trial less than6×10?3).The horizontal bars on the frequency points indicate the extent of the interval for which the power density spectrum was calculated.In a few cases,more than one peak exceeds this threshold in a given power density spectrum.We then represent the lower-power peak by an open circle.In burst a the secondary peak has a Leahy power of21.2 (single-trial signi?cance2.5×10?5);in burst b the secondary peaks have Leahy powers of44.0 (?rst interval;signi?cance2.8×10?10)and13.5(second interval;signi?cance1.2×10?3);and in burst c the secondary peak has a Leahy power of12.8(signi?cance1.7×10?3).Finally,Figure3 shows the rms amplitude of each oscillation,computed for one-second intervals1/8second apart.

It is evident from these?gures that the frequency behavior can be very complex and can

di?er greatly from burst to burst.The light curves for bursts(a),(c),and(d)appear similar to each other,although burst(d)has a slightly longer decay time than the other two.However,the frequency and amplitude of the brightness oscillations evolve very di?erently in the three bursts.

In burst(a),there is a strong oscillation near the beginning which disappears for approximately one second,then the oscillation reappears after the peak.The frequency increases continuously, although there is some evidence that in the initial~0.5second of the burst the frequency drops (this might help explain the presence of a higher-frequency secondary peak in the power density spectrum).

In burst(c),the brightness oscillation is present for almost the entire time examined.The frequency increases rapidly in the?rst two to three seconds,then appears to decrease to an asymptotic value.A power density spectrum of a two-second interval starting1.75seconds after the beginning of the burst reveals a peak at581.62±0.04Hz.A power density spectrum of a six-second interval starting four seconds after the beginning of the burst has a peak at 581.47±0.01Hz.If the latter frequency is the asymptotic frequency of the oscillation,then at a3σlevel of certainty it is less than the maximum frequency attained during the burst.The amplitude of the oscillation in the burst tail is high and signi?cant,and there is an abrupt increase in the amplitude6–8seconds after the beginning of the burst that is not accompanied by any apparent change in the light curve.

In burst(d)there is a clear decrease in the frequency of the burst oscillation in the tail of the burst.We explored this further by taking a power density spectrum of a longer interval:?ve seconds,starting three seconds after the beginning of the burst.We found that,at the99.99% con?dence level,the frequency change per second during this interval is?0.54±0.08Hz s?1.The best-?t frequency at the beginning of this?ve-second interval depends on the frequency derivative,

Fig. 2.—Power spectra as a function of time for the four bursts with strong brightness oscillations.Each solid triangle is at the frequency of maximum power and its1σuncertainties for nonoverlapping1second intervals.The frequency of a peak is only plotted if its Leahy power exceeds10.In bursts1,2,and3there are intervals in which a second peak exceeds this threshold, and this secondary peak is plotted with an open circle.The power of the secondary peak in burst(a) is21.2(single-trial signi?cance2.5×10?5);the powers of the secondary peaks in burst(b)are44.0 (signi?cance2.8×10?10)and13.5(signi?cance1.2×10?3);and the power of the peak in burst(c) is12.8(signi?cance1.7×10?3).The panels are labeled as in Figure1.Burst(e)does not have

strong brightness oscillations for most of the burst,and is therefore excluded from this analysis.

Fig. 3.—Root mean square amplitude of the brightness oscillation for each of the bursts.The amplitudes are calculated for one-second intervals with starting times1/8second apart,and the ±1σuncertainty bands are shown.The amplitude is only plotted if the Leahy power for the

oscillation exceeds10.The panels are labeled as in Figure1.

and is approximatelyν0=581.39Hz?2(˙ν+0.62Hz s?1)Hz.This means that,relative to a brightness oscillation with a constant frequency equal to the frequency at the beginning of this ?ve-second interval,the observed brightness oscillation has a total phase lag of between12πand 16πradians.The total phase lag is comparable to what is seen in many bursts,except that here the frequency inferred in the tail of the burst is signi?cantly less than the spin frequency inferred from other bursts in this source.There is no sign in this burst that the frequency has reached an asymptotic value.

Burst(b)is the only one with a clearly di?erent light curve.This is a weak burst.The frequency of the brightness oscillation is consistent with what is observed in,at least,bursts(a) and(c):a rise in the frequency near the beginning of the burst,followed by an approximate leveling o?.We note,however,that within the uncertainties the frequency could also reach a maximum and then decline,as appears to be the case for bursts(c)and(d).

3.BRIGHTNESS OSCILLATIONS AT THE BEGINNING OF THE BURST

Previous analyses have shown that the brightness oscillations in the initial~second of

the bursts are often of particular interest.This is where the highest amplitudes(rms~50%; Strohmayer et al.1998c)are reported,and where subharmonics of the strong oscillation have been detected in4U1636–536(Miller1999)and possibly in the Rapid Burster(Fox&Lewin1999). It is therefore important to examine the initial portion more closely to see what hints about the brightness oscillation mechanism can be derived.

Before doing so,we need to emphasize an important distinction.If the purpose of the analysis is to characterize the frequency variations of the~580Hz oscillation then extra parameters can only be added if the?t to the data is improved su?ciently to justify the additional complexity.The situation is di?erent when the goal is to produce a matched?lter for a search for a harmonically related frequency,as in the search for a signal at half of the~580Hz dominant brightness oscillation in4U1636–536(Miller1999).For that purpose,it is not necessary to justify the extra parameters used in the construction of the?lter,if no reference is made to the signal for which one is searching.In the case of the search for the~290Hz oscillation,a?ve-parameter matched?lter was used for each burst;matched?lters with fewer parameters also give a clear signal at~290Hz, although with lower signi?cance because the?lter does not?t the data as well.

A general method to?nd the best-?t values of parameters and their con?dence regions employs a likelihood function.In this approach,we suppose that we have a model in which the countrate as a function of time is predicted to be s(t),from which we can predict the number of counts s i in one particular bin i of the data,which in this case is1/8192s in duration.In general, s i is not an integer.The actual number of counts observed in bin i is c i,which is an integer.With

these de?nitions,the Poisson likelihood of the full data set given the model s(t)is

s c i i

L=Π

If the amplitude A?1,as it is in this case,then a tremendous speed-up in the search procedure is possible with the use of the cross-correlation(see,e.g.,Helstrom1960or Wainstein&Zubakov 1962for details of cross-correlation and matched?ltering techniques)

H=C t0+T t0c(t)e?iν(t)t dt

2,(7)

where t0is the start time of the burst,T=0.75s is the duration of the burst,and C is a normalization constant.In practice this integral is actually calculated as a sum over all of the bins of the data,and dt=1/8192s is the duration of a bin.If C=2/N tot,where N tot is the total number of counts in the data set,then H has the same statistical properties as the Leahy power;

H is also related to the Z2statistic used in pulsar period searches(Buccheri et al.1983;see Strohmayer&Markwardt1999for a recent use in the characterization of brightness oscillations during thermonuclear X-ray bursts).To lowest order in the oscillation amplitude A this description is mathematically identical to the likelihood description,but it is much faster to apply because no search need be performed for the amplitude or oscillation phase.It is therefore preferable for low-amplitude oscillations.

With this formalism,we can estimate the best values and uncertainty regions for the di?erent frequency models above.The?gures in the previous section,which were constructed using a constant-frequency waveform,give this information for the one-parameter,constant-frequency model.

3.1.Two-Parameter Frequency Model

The best-?t values for the two-parameter frequency model are given in Table2.To estimate uncertainties on these parameters,we performed a Monte Carlo analysis in which we selected106 random values per burst ofν1,˙ν1,˙ν2,and t break,uniformly sampled from,respectively,576Hz to585Hz;-12Hz s?1to12Hz s?1;-12Hz s?1to12Hz s?1;and0s to0.75s.The quoted uncertainties for single parameters were computed using a Bayesian viewpoint,in which the posterior probability density was calculated throughout the interval and then integrated over the other three parameters to produce a marginalized probability distribution.We have assumed a uniform prior probability density over the whole space searched.This means that the posterior probability density is simply proportional to the likelihood.These con?dence regions,which are the smallest regions that encompass68%of the probability,are given in Table3.In some cases the maximum likelihood value of a parameter obtained by extremization in the full two-dimensional parameter space is outside the marginalized68%con?dence region.This is symptomatic of the fact that the parameters are constrained only weakly by the data.

Table2:Best-Fit Parameters

for Two-Parameter Model

Burstν1(Hz)˙ν1(Hz s?1)

Table3:68%Con?dence Regions

for Two-Parameter Model

Burstν1(Hz)˙ν1(Hz s?1)

Table4:Best-Fit Parameters for Four-Parameter Model

Burstν1(Hz)˙ν1(Hz s?1)˙ν2(Hz s?1)t break(s)

3.2.Four-Parameter Frequency Model

The best-?t values for the four-parameter frequency model are given in Table4.As for the two-parameter model,the uncertainties were estimated by marginalizing over all but the parameter of interest;the con?dence regions containing68%of the probability are given in Table5.

3.3.Five-Parameter Frequency Model

The best-?t values for the?ve-parameter frequency model are given in Table6.As for the two-parameter model,the uncertainties were estimated by marginalizing over all but the parameter of interest;the con?dence regions containing68%of the probability are given in Table7.

Table5:68%Con?dence Regions for Four-Parameter Model

Burstν1(Hz)˙ν1(Hz s?1)˙ν2(Hz s?1)t break(s)

Table6:Best-Fit Parameters for Five-Parameter Model

Burstν1(Hz)ν2(Hz)˙ν1(Hz s?1)˙ν2(Hz s?1)t break(s)

Table7:68%Con?dence Regions for Five-Parameter Model

Burstν1(Hz)ν2(Hz)˙ν1(Hz s?1)˙ν2(Hz s?1)t break(s)

3.4.Summary of Frequency Models

The best-?t parameters and relative log likelihoods are listed in Table8;as indicated above, 2?log L≈?χ2.From this table,it is clear that for all but burst four it is not necessary to use the ?ve-parameter?t,and for bursts2,3,and4it is not necessary to use a model more complicated than the two-parameter model in which the frequency and frequency derivative are constant throughout the?rst0.75seconds.For burst(d)by itself the?ve-parameter model is preferred at only the2σlevel compared to the four-parameter model,and for all?ve bursts combined the ?ve-parameter model is preferred at less than the1σlevel relative to the four-parameter model. For all?ve bursts combined,the four-parameter model is preferred at less than the1σlevel compared to the two-parameter model,and the two-parameter model is preferred at less than the 2σlevel compared to the one-parameter model.

Table8:Relative Log Likelihoods for Di?erent Models

Burst1-Param2-Param4-Param5-Param

4.DISCUSSION AND SUMMARY

What can be learned from this detailed characterization of the burst brightness oscillations in4U1636–536?The clearest impression left is that there are no simple statements about the frequency behavior that are true for all of the bursts.In two of the bursts,one can make an argument that the oscillation frequency is initially1–2Hz below the asymptotic frequency,and then rises.In this interpretation,the asymptotic frequency is extremely close to the spin frequency of the neutron star.This picture can be qualitatively explained by the idea that the burning layer lifts20–50meters during the burst and settles down gradually.However,the burst on31December 1996does not follow this pattern.The frequency in the initial second is indeed lower than the maximum value attained,but the signi?cance of this initial signal is low(Leahy power of10). The maximum is followed by a clear decrease in the frequency over several seconds,with a total phase change equivalent to more than?ve complete circuits around the star.This happens during a time when the countrate decreases from approximately2/3of the maximum to approximately 1/3of the maximum.The burst on29December1996has a very strong and signi?cant brightness oscillation in its tail,which appears to level out to a constant frequency.However,near the peak of the light curve for this burst the oscillation frequency is higher than this asymptotic frequency, at a3σsigni?cance level.

Such a drop in frequency is not expected in the simplest version of the hypothesis that the frequency changes are caused by the rise of the burning layers.In this model,the highest frequency should be observed when the layers are fully coupled to the core of the star,which is expected to occur when the frequency has reached its asymptotic limit.

Another constraint on the hypothesis that the asymptotic frequency equals the spin frequency (after correcting for orbital Doppler shifts)is that the variation in the observed asymptotic frequency must be consistent with the possible modulation due to the binary motion of the neutron star.From binary evolution theory(see,e.g.,Lamb&Melia1987;Verbunt&van den Heuvel1995),an LMXB such as4U1636–536with a3.8hr orbital period(van Paradijs et al. 1990)that contains a neutron star of mass M NS=1.4M⊙to2.0M⊙has a companion star of mass M c≈0.4M⊙.Assuming that the orbit is approximately circular,the orbital velocity of the neutron star is therefore90–130km s?1,implying a maximum frequency modulation of?ν/ν=4.3×10?4, or approximately0.25Hz ifν=580Hz.Therefore,the observed asymptotic frequency cannot be

di?erent by more than0.5Hz for two di?erent bursts.The analysis of the31December1996 burst reported in§2indicates that eight seconds after the start of the burst the frequency is less than579.0Hz.The asymptotic frequency in the burst on29December1996is581.43Hz,so the frequency in the31December1996burst must rise by2Hz to reach a plausible spin frequency.

It is di?cult to reconcile this frequency behavior with what is expected in the simplest version of the rising burning layer hypothesis.One possibility is that the observed frequency changes are not simply indicative of the spin frequency of the burning layer,but also include a time-dependent change in the phase at which the photons emerge relative to the phase of the burning layer.This would be observationally indistinguishable from a pure frequency change,and would add an extra degree of freedom to the model.

Even this,however,is subject to signi?cant observational restrictions.To see this,consider the following observational trends,which have been observed in many bursts from several sources (see,e.g.,Strohmayer et al.1998for a summary).In the remainder of this section we assume that all quantities(e.g.,frequencies,times,and phases)are measured at in?nity.

(1)There are several bursts in which burst oscillations are seen for the entire burst,and do not disappear during the time of peak countrate.

(2)Aside from an early phase in which there may be a frequency decrease,the frequency increases smoothly as the burst progresses.

(3)The total phase lag of the oscillations compared with a hypothetical oscillation that has a constant frequency equal to the frequency in the burst tail is as much as10π.

The total amount of energy in a burst is~1039ergs.If expansion of a layer and angular momentum conservation are to explain the~0.3%–1%change in the observed angular frequency, then the layer must rise by a distance that is a fraction~0.2%–0.5%of the radius of the star,or 20to50meters.The surface gravity of a neutron star is~2×1014cm s?2,so the largest amount of mass that can be lifted to the required20–50meter height above the surface is~1–2×1021g.If most of the~1013cm2surface area of the star is involved,this implies that the greatest column depth which could be lifted to the required height is roughly108g cm?2,which is comparable to the expected106?108g cm?2depth of ignition(see,e.g.,Fushiki&Lamb1987;Brown&Bildsten 1998).

One may therefore distinguish two scenarios:(1)the burning layer rotates with the core of the star at a constant spin frequency and the observed frequency shifts are caused by phase shifts induced by radiation transport through more slowly rotating layers,and(2)the burning layer itself is lifted and rotates more slowly than the core of the star.We now treat these in order.

Suppose for simplicity that the burning layer has an in?nitesimal vertical extent,that it has some restricted azimuthal extent,and that it all rotates with the same angular frequencyωburn(t). The energy from this layer propagates upwards through the atmosphere,which in general may be composed of layers with di?erent angular frequencies.Therefore,the phase of emergence of

the radiation may di?er from the phase of the burning layer at the time of the emission of the radiation.Under the rising burning layer hypothesis,it is expected that the angular frequency of higher layers is less than the angular frequency of lower layers(dω/dh<0).Hence,there is expected to be a lagφlag>0between the phase of emergence and the phase of emission.This phase lag will,in general,have a time-dependence,as the scale height of the atmosphere and the angular frequency of di?erent layers in the atmosphere changes throughout the burst.An observer at in?nity will therefore see a net angular frequency of a hot spot that is equal toωburn(t)?˙φlag(t).

Consider?rst a burning layer that rotates with the stellar core throughout the burst.Then ωburn(t)=const=ωspin.If neitherω(h)nor the density or height of the envelope changes with time,thenφlag is a constant and the observed frequency is justωspin.Hence,in order to have an apparent frequency shift in this situation,the structure or angular velocity of the envelope must change with time.

Now consider an envelope that does change with time.For us to observe a frequency less than ωspin,it is necessary that˙φlag(t)>0,so the characteristic phase of emergence of the radiation must lag the phase of the source of heat by a greater and greater amount with increasing time (the increase of this phase lag with time must itself decrease with time to produce the observed increase in frequency).But how is this possible?As the envelope settles down,the phase lag should decrease,because dω(t)/dh<0.But if the phase lag decreases,the observed frequency should be higher than the spin frequency.This is not seen in most bursts,and even in the burst on29December1996where there does appear to be a short period of spindown,the total phase lead implied by the spindown is much smaller than the total phase lag implied by the spinup near the beginning of the burst.Thus,the preceding set of assumptions is inconsistent with the data.

This demonstrates that the observed frequency behavior is inconsistent with the source of heat(i.e.,the burning layer)rotating at a constant frequency equal toωspin.Instead,the source of heat must change its frequency during the burst.

To analyze this situation,let us now consider a burning layer with a?nite thickness,so that the observed photons are a superposition of the photons from many in?nitesimal layers such as discussed above.The observed frequency of oscillation is then a superposition of the frequencies due to the in?nitesimal layers.

Consider two of these in?nitesimal slices,labeled1and2,where slice1is higher than slice 2.Suppose that these slices are not coupled to each other.Then,by assumption,the angular frequencyωburn,1of slice1is less than the angular frequencyωburn,2of slice2.In addition,because the photons from slice2have to travel through the same atmospheric layers as the photons from slice1in addition to the layers between2and1,the phase lagφlag,1of photons from slice1is expected to be less than the phase lagφlag,2of photons from slice2.Hence,as the atmospheric scale height decreases,it is expected thatφlag,2will decrease more rapidly thanφlag,1does,so that

˙φlag,2<˙φ

lag,1<0.(8)

Therefore,the di?erence between the angular frequency of the photons from slice2and the angular frequency of the photons from slice1is

ωburn,2?ωburn,1+˙φlag,1?˙φlag,2>ωburn,2?ωburn,1.(9) This means that the phases of emergence of radiation diverge rapidly from each other,which leads quickly to a low amplitude unless the heat source has a small vertical extent.The requirement that the amplitude be signi?cant means that the total azimuthal phase subtended by the emergent radiation has to be much less than2π.The integrated phase lag relative to the stellar core is often 10πor larger,hence the average vertical extent of the heat source must be much less than1/5of the vertical distance from the original location of the heat source to its location during the burst. An alternative to having the vertical extent of the layer be small is that the burning layer may be tightly coupled to itself,so that its angular frequency is approximately constant over a signi?cant vertical distance.

To summarize,several conclusions may be drawn about the standard model for frequency changes during burst oscillations,which we take to be the picture that at least part of the burning layer is lifted and then settles gradually to the surface as the?ux drops,producing an observed asymptotic frequency equal to the spin frequency of the neutron star Doppler-shifted by the orbital motion of the neutron star.(1)The burning region itself(and not just overlaying optically thick layers)must be lifted by20–50meters from the surface,(2)this region must remain decoupled from the rest of the star,presumed to be rotating at the original spin frequency, for several seconds,(3)to produce the observed coherence of the brightness oscillations during the rise in frequency,the burning layer must either have a vertical extent much smaller than its height above the surface or be strongly coupled to itself to prevent relative azimuthal motion,and (4)the existence of a frequency greater than the asymptotic frequency(as in the29December 1996burst)implies that something other than di?erential rotation(e.g.,variation in the phase lag)must account for at least part of the observed frequency change.The prolonged decrease in frequency in the tail of the31December1996burst is not straightforwardly?t into this picture.

Despite these di?culties,the high stability(Strohmayer et al.1998)and coherence(Markwardt &Strohmayer1999)of the brightness oscillations in the tails of bursts from sources such as

4U1728–34argue persuasively that the frequency in the tail of the bursts is close to either the fundamental or the?rst overtone of the neutron star spin frequency.Moreover,the general picture in which frequency changes are attributed to changes in the height of the emitting layer accounts approximately for the magnitude of the frequency change and explains why the frequency tends to rise near the beginning of the burst.However,in its current form it su?ers from apparently serious problems.It is extremely important that there be a detailed investigation of,e.g.,the coupling between di?erentially rotating layers,and that other ideas be explored so that the strengths and weaknesses of the rising layer model are put into sharper focus.

We thank Don Lamb and Fred Lamb for discussions about models of the frequency change,

and Don Lamb,Dimitrios Psaltis,and Carlo Graziani for comments on a previous version of this paper.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. This work was supported in part by NASA grant NAG5-2868,NASA AXAF contract SV464006, and NASA ATP grant number NRA-98-03-ATP-028.

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