Why is it Difficult to Detect a Millisecond Pulsar in Neutron Star X-ray Binaries?
Lev Titarchuk \altaffilmark{1,3}
Wei Cui\altaffilmark{2} and Kent Wood \altaffilmark{3}
We explain why it is possible to detect directly X-ray emission from near the surface of the neutron star (NS) in SAX J1808.4-3658 but not in most other low-mass X-ray binaries (LMXBs), with the exception that emission from the surface can be seen during bursts events. We show that the X-ray emission from SAX J1808.4-3658 mostly originates in the Comptonization process in a relatively optical thin hot region (with an optical depth $\tau_0$ around 4 and temperature is around 20 keV). Such a transparent region does not prevent us from detecting coherent X-ray pulsation due to hot spots on the NS surface. %When the %characteristic scattering time in the region is of order of inverse %rotational frequency $\omega$ or higher then the amplitude of the %harmonic oscillation decays exponentially with $\tau_0\omega$. We give a precise model for the loss of modulation, such suppression of the QPO amplitude due to scattering can explain the disappearance of kHz QPOs with increasing QPO frequency. We also formulate general conditions under which the millisecond X-ray pulsation can be detected in LMXBs. %We argue that kHz QPOs observed in %LMXBs are associated with the resonance effects. %We show that the accuracy %of the eigenfrequency extraction using the resonance frequencies is no %better than a few percents in contrast to the previous claims. We demonstrate that the observed soft phase lag of the pulsed emission is a result of the downscattering of the hard X-ray photons in the relatively cold material near the NS surface. In the framework of this downscattering model we propose a method to determine the atmosphere density in that region from soft-lag measurements.
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%\received{ 2002  May 7 } 
%\accepted{2002 July 24 by the Astrophysical Journal Letters} 
%\journalid{337}{1 January 1995} 
%\articleid{11}{14} 
 
\slugcomment{Accepted for publication in {\it the Astrophysical Journal Letters} 2002 July 24 } 
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\begin{document} 
 
\title{Why is it Difficult to Detect  a Millisecond Pulsar in Neutron Star 
X-ray Binaries?} 

\author{Lev Titarchuk \altaffilmark{1,3}, Wei Cui\altaffilmark{2}
and Kent Wood \altaffilmark{3}} 


\altaffiltext{1}{George Mason University/Center for Earth Observing and
Space Research, Fairfax VA 22030-4444}
\altaffiltext{2}{Department of Physics, Purdue University, West Lafayette,
IN 47907; cui@physics.purdue.edu}
\altaffiltext{3}{US Naval Research Laboratory,  Space Science
Division, 4555 Overlook Avenue, SW, Washington,
DC 20375-5352; lev@xip.nrl.navy.mil; kwood@ssd5.nrl.navy.mil}

 
 
\vskip 0.5 truecm 
 
%c\font\rom=cmr10 
%c\centerline{\rom Accepted, Astrophys. J. } 
 
 
\begin{abstract} 
We explain why it is possible to detect directly X-ray emission from near the 
surface of the neutron star (NS) in SAX J1808.4-3658 but not in most other
low-mass X-ray binaries (LMXBs), with the exception that 
 emission from the surface can  be  seen 
during bursts events. We show that the X-ray emission from 
SAX J1808.4-3658 mostly originates in the Comptonization process 
in a relatively optical thin hot region (with an optical depth $\tau_0$ 
around 4 and temperature is around 20 keV). Such a transparent region 
does not prevent us from detecting coherent X-ray pulsation due to hot 
spots on the NS surface. 
%When the 
%characteristic scattering time in the region is of order of inverse 
%rotational frequency $\omega$ or higher then the amplitude of the 
%harmonic oscillation decays exponentially with $\tau_0\omega$. 
We give a precise model for the loss of modulation,
such suppression of the QPO amplitude due to scattering can explain
the disappearance of kHz QPOs with increasing QPO frequency. 
We also formulate general conditions under which the millisecond X-ray 
pulsation can be detected in LMXBs. 
%We argue that kHz QPOs observed in 
%LMXBs are associated with the resonance effects. 
%We show that the accuracy 
%of the eigenfrequency extraction using the resonance frequencies is no
%better than a few percents in contrast to the previous claims. 
We demonstrate that the observed soft phase lag of the pulsed emission
is a result of the downscattering of the hard X-ray photons in the 
relatively cold material near the NS surface.
In the framework of this downscattering model we propose 
a method to determine the atmosphere density in that  region from soft-lag
measurements.
\end{abstract} 
 
%\keywords{X-rays: bursts-stars --- accretion disks, LMXB, X-ray 
%binary --stars: neutron ---stars: 
%individual: 4U 1728-34,  GX 340+0, Sco X-1} 
 
\keywords{accretion, accretion disks --- pulsars: individual (SAX
J1808.4-3658)--- stars: neutron ---X-rays: stars} 
 
\section{Introduction} 
Low-mass X-ray binaries (LMXB) presumably contain 
a  weakly magnetized neutron star (NS).
Recently Titarchuk, Bradshaw \& Wood (2001), hereafter TBW 
suggested a new method of estimating B-field strength  using
the magnetoacoustic oscillation model and  found that 
the B-field strength for a number 
of NSs in  LMXBs is  $\sim 10^8$ gauss.  
%They showed that the model fit to the observable correlation
%between the viscous frequencies and low kHz QPO frequencies leads to estimate of the NS B-field
%of $10^{7-8}$ Gauss in 4U 1728-34, Sco X-1 and GX 340+0.
Near-coherent millisecond X-ray pulsations have been 
observed in 4U 1728-34
during thermonuclear (type I) X-ray bursts (e.g. Strohmayer et al. 1996).
They are interpreted as X-ray intensity  modulated at period close to the spin period of the neutron star. 
Thus these B-field estimates in addition to the detection of the milisecond pulsation 
give strong arguments for LMXB neutron stars to be   progenitors of millisecond radio pulsars (MLP)
(see review by Bhattacharya \& van den Heuvel 1991). 
 But there is still a question why these coherent pulsations are not found in persistent emission
 despite careful searches (Wood et al. 1991; Vaughan et al. 1994).
The lack of coherent pulsations has been explained as modulation loss from 
 gravitational lensing (Wood, Ftaclas \& Kearney 1988; Meszaros, Riffert \&
Berthiaume 1988) or from to scattering (e.g. Bainerd \& Lamb 1987; Kylafis \& Klimmis 1987 ).
% Wang \& Shlickeiser 1987; and
%Bussard et al. 1988).  
The third explanation  for  the lack  of pulsations is presented by 
Cumming, Zweibel \& Bildsten (2001) who argue that the surface field is weak because of magnetic
screening  \footnote{%It is consistent with the fact that SAX 1808.4-3658, has an unusually low
%accretion rate of $\dot M\sim 10^{-3}M_{\rm Edd}$ 
The recent discoveries of MLP in XTE J1751-305 (Markwardt et al. 2002) and in 
XTE J0929-314 (Galloway et al. 2002 ) with extremely low mass transfer rates support the suggestion 
of the absence  of the magnetic screening  for the  low mass accretion rates.}.
In this {\it Letter} we put forth arguments for smearing out of the pulsar signal due to 
electron scattering in the optically thick environment typical of  most  observed 
LMXBs.
% \footnote{%The observed X-ray spectra in Sco X-1, Cyg X-2, GX340+0 and 4U 1728-34 are 
%characterized by a Comptonized Blackbody formed in the optically thick cloud with an optical depth
%higher than 5 (Titarchuk et al. 2001, TBW).
% and Kuznetsov \& Titarchuk 2002a,hereafter KT02).
%If this Compton cloud completely covers the NS surface it is not surprising that the pulsar signal 
%is completely smeared out. 
%The resulting  time signal as a result of the NS pulsation followed
%by scattering is a convolution of the pulsar signal and the response of the scattering in  the
%cloud (the Green's function).}.  
In  \S 2 we study the scattering effect and its relation with
the observed timing and spectral characteristics  for various QPO sources.
In \S 3  we investigate the scattering effects in QPO sources.
In \S 4 we present results of the RXTE data analysis of spectral  properties of X-ray radiation in   
SAX 1808.4-3656. In \S 5 we analyze the downscattering model and its application to the observed
soft lag phenomenon detected in the coherent pulse signal from SAX 1808.4-3656.
Conclusions follow in \S 6. 

\section{Scattering Effect on  QPO Pulse Profile and Pulsar Profile}  

The transfer problem to be solved is that of a sinusoid, representing either (a)
an intrinsic pulsar light curve or (b) a high frequency QPO, but in each case
seen through a surrounding medium that reduces the pulse modulation 
by scattering. The signal whose modulation is affected lies in the range
of a few hundred Hz to perhaps 1 kHz. We thus must analyze smearing out of a signal
at very high frequencies and investigate the dependence upon both the intrinsic
frequency and the optical depth of the scattering environment. The geometry used
derives from earlier works (see e.g. Sunyaev \& Titarchuk 1980, hereafter ST80;
Titarchuk, Lapidus \& Muslimov 1998, hereafter TLM98)
% Titarchuk, Osherovich \&
%Kuznetsov 1999, hereafter TOK) 
modeling energy spectra and power density
spectra (PDS) of LMXB sources. 
%These models give a picture of energy release
%that singles out a particular radius where temperature climbs steeply and plasma
%is removed from the disk to form a corona. 
{\it For the high frequency QPO,  [case (b)]
the TLM98 model identifies the site where the QPO is generated}, which will be at radius substantially
larger than that of the star. Thus the effective optical depth of interest in case (b) will not be
the full optical depth to the star surface but rather the reduced optical depth appropriate to that
site location. 

% Cutting is done 07/10/02
% TLM98 considered the possibility of dynamical adjustments of a Keplerian
%disk to the innermost sub-Keplerian boundary conditions. Their picture is explains
%most observed QPO phenomenology  in bright LMXBs with minimal parametrization, which in turn gives
%confidence in the geometry that is inferred. High frequency QPO effects are particularly associated
%with the transition radius where flow becomes non-Keplerian. 
%An isothermal sub-Keplerian transition  layer
%between the NS surface and its last Keplerian orbit forms as a result
%of this adjustment.  The TLM98  model for the PDS features therefore provides the needed
%information on the QPO site and its relation to the scattering medium.

%TLM98 showed how  the discontinuities and abrupt
%transitions in their solution result from derivatives of quantities such as angular velocities
%(weak shocks). Subsequently, Titarchuk \& Osherovich (1999 hereafter,
%TO99) identified the model's
%viscous frequency and diffusion  frequency, the latter corresponding to a break in the PDS.
%from the model, kHz QPOs are generated at radii of roughly 20 km; one QPO frequency is 
%in fact just the Kepler frequency at this frequency.

% cutting is done in 07/11/02
%In TLM98 the transition layer (TL) outer boundary, at the adjusted radius (see Fig. 1)
%  is presumably the place where the 
%incoming accretion flow release its gravitational energy. 
%Because the flow releases its energy
%in the optical thin target the temperature of plasma in this particular place can be much higher
%than that in the disk (TLM98). 
%This gives hotter plasma in a corona above the disk, flowing to the neutron star at the poles
%while cooler material in deeper disk layers proceeds towards the neutron star in the disk plane.
%The model of the energy and power spectra then predicts the optical depth from QPO site to the
%observer $\tau_0$ to have value of $\tau$ about twice less than the optical depth $\tau$ 
%from the NS star surface to the observer. Below we show that in SAX J1808.4-3658 
%the best-fit parameters  $\tau_0$ and
%$\tau$ are $\sim 2.7$ and $\sim 4.5$ respectively.

One more consideration completes the problem specification.
It is well established from observation that the kHz frequency QPOs in LMXBs are not fixed in frequency
but increase or decrease in manner correlated with changes in luminosity.
which in turn is believed
to vary directly as mass accretion rate, hence as source brightens the OPO site moves both inward in
radius and upward in frequency. 
Consequently, 
the total change in QPO power as the source varies
will be a combination of two effects, namely (i) the intrinsic change in QPO power that results
simply from physics of shifting the resonance frequency that gives the QPO and (ii) the further
reduction in power that results from its now being {\it observed at higher frequency and through
 greater optical depth}, both of which enhance the effects of scattering.  
To model the entire process we therefore require a model for the intrinsic change in power.
This is provided in the cited transition layer (TL) model (TLM98)
% Titarchuk \& Osherovich 1999, hereafter,
%TO99) 
by approximating the QPO process itself as
a damped harmonic oscillator under the influence of a driving force. Then, the interaction leads to
resonance oscillations for which the Q-value $(\nu_{QPO}/\delta\nu_{QPO})$ can be high.
The resonance amplitude varies as the product of the amplitude of the driving force $A_{dr}(\nu)$ and
$1/\nu_{QPO}$ (see Landau \& Lifshitz 1965). The default intrinsic model, in the absence
of scattering, is thus that the QPO amplitude  falls off as  $A(\nu)=A_{dr}(\nu)/\nu_{QPO}$, but scattering
contributes additional reduction. In a source such as GX 340+0, where a {\it pair} of high frequency
QPOs is seen, it is possible to test out these ideas in detail, using the model derived below and
applying it to both QPOs, keeping the optical depth $\tau_0$ the same for both kHz QPOs because they
are co-located.
Detection of the NS spin period is the crucial further issue at stake here.
% cutting is done in 07/11/02
%The models cited above require
%a magnetosphere to break the symmetry of the problem and create some of the observed QPOs.
%When the full complement of 6 QPOs and break frequencies predicted by the TL model are observed it
%becomes possible to estimate the NS surface magnetic field strength, or equivalently its dipole moment
%(TBW). The result is an estimated surface field in the range $10^7-10^8$ gauss, which should in
%principle suffice to produce periodicity in the emission at the spin frequency. An evolutionary link to
%radio millisecond pulsars requires a similar value.  We know of a few cases where spin period
%modulation is detectable, but these are exceptions to the rule.
The typical problem is the following: since the NS spin gives a {\it coherent} periodic signal at
nearly constant frequency 
%(with allowance for drifts either during X-ray bursts or associated
% with the Doppler effect of its orbital motion and perhaps accretion torques) 
it should be at relative
 {\it advantage}  in detectability in comparison with QPOs, yet is not see while the QPOs {\it are
 seen}. 
 %\footnote{
 %A model without scattering has little hope of handling this full problem. 
 %The ultimate test of
 %a scattering picture, then, is its ability to account for the general absence of detections at stellar
 %spin frequency in a manner consistent with simultaneous detection of the QPOs that are produced
 %nearby both in physical space and frequency space}. 
 The key, obviously, is that the QPO sees the
 reduced optical depth while the modulation at the spin period experiences the unreduced optical depth.
 The occasional exceptions where the stellar spin frequency does prove detectable (such as MLPs,
  plus the transient detections in X-ray bursters) should be explicable in terms of
 circumstances that reduce scattering effects in those particular cases below values that generally
 obtain; moreover those exceptional reductions should be confirmed by spectra indicative of the lower
 optical depth and  other supporting context observations.
 %The streams of the hot matter bouncing from the disk 
%converge towards the NS along magnetic lines (see Fig. 1).
%On the other hand the flow of the deeper 
%disk layers which are much cooler continue to proceed towards NS being in the disk plane. The hot streams of plasma forming a corona above the disk (Compton cloud) creates 
%the medium through which an observer  has to see the NS surface.  The hot plasma flow heats
%the NS  at the magnetic poles making the hot spots on the NS surface.
%In Figure  1 we present a possible  geometry of the source SAX 1808.4-3656. 

For a quantitative model of this problem scattering  is handled as a diffusion problem in a Green's
function [$G(t)$] treatment.
The input  pulse is $x(t)=A\sin({\omega
t}+\varphi_0)$ where $\omega=2\pi \nu$ is the NS spin rotational  frequency $\omega$ or the QPO 
centroid frequency depending on whether it case (a) or (b) and 
$\varphi_0$ is an initial phase.
%Another source of harmonic pulsations is a harmonic oscillator under the influence 
%of  a driving force. This interaction leads to the resonance oscillations for which 
%Q-value ($\nu/\delta \nu$) also can be  very high. 
% The disk and NS rotations and the broadband variability (which can be generated by diffusion processes in 
%the transition layer, see TO99 and TLM98),   might be a source of such driving 
%forces. It is well known (see Landau
%\& Lifshitz 1965, hereafter LL) that 
%the resonance amplitude is a product of the amplitude of the driving
%force [which is a square root of the power density (PDS)  at the given frequency] and 
%$\omega^{-1}$ .
%The pulsed emission propagating  outward is affected by scattering 
%in Compton cloud.
The resulting pulse affected by the scattering is $z(t)$ and is given by a convolution of $x(t)$ and 
$G(t)$:
\begin{equation}
 z(t)=\int_0^t x(t^{\prime})G(t-t^{\prime})dt^{\prime}.
\end{equation}   
For times $t$ much greater than the scattering time in the Compton cloud 
$R\tau/c$ the initial pulse emission $x(t)=A(\omega)\exp{[(\omega t+\varphi_0)i]}$ is
calculated as follows
\begin{equation}
z(t)=A(\omega)\exp{[i(\omega t+\varphi_0)]}I(\omega),~~~~
%\end{equation}
{\rm and}  ~~~~
%\begin{equation}
I(\omega)=\int_0^{\infty}e^{-i\omega u}G(u)du,~~~
\end{equation}
ST80
analyzed the response of the scattering medium (the Green's function)
for various source distributions. 
They show that
for a QPO source embedded in the center of the cloud ($\tau=0$)
the response of the scattering medium (the Green's function), $G(t)$ 
can be approximated by 
\begin{equation}
 G(t)\propto(C_1+C_2t^{-5/2})\exp[
 -3R\tau_0/4ct
 -\pi^2c\tau_0t/3R(\tau_0+2/3)^2],
 \end{equation}
 where $\tau_0$, $R$ are  optical depth and radius of the cloud respectively,
 $c$ is the light speed. Coefficients $C_1$ and $C_2$ can be estimated as
 follows (see also Wood et al. 2001)
% \begin{equation}
 $C_1=5\pi^2/3(\tau_0+2/3)^2$, 
% \end{equation}
and 
%\begin{equation}
 $C_2=\pi(3/\pi)^{3/2}\tau_0^{1/2}
 (R/c)^{5/2}/4$.
 %\end{equation}
Using formula (3) the integral  $I(\omega)$  is analytically calculated  
 by the steepest descent method 
 %or by deriving the appropriate first order differential equation
 %(this is the case for the second term  with $t^{-5/2}$ of the sum of  Eq. 3,
 (see also  Prudnikov, Bruchkov \& Marichev 1981, formula 2.3.16).
 We can show  that an amplitude $B(\omega)$ of $z(t)$
 can be obtained as
 \begin{equation}
 B(\omega)= A(\omega){\cal F}({\cal X})
 \exp{ (-2\cal X)}
 \end{equation}  
 where ${\cal X}(\omega,\tau_0)=\sqrt{[\pi\tau_0/2(\tau_0+2/3)]^4+(3R\tau_0 
 \omega/4c)^2}$ and ${\cal F}({\cal X})=d_1{\cal X}^{1/2}+
 d_2{\cal X}^{-3/4}+d_0$ is a weak function of
 ${\cal X}$
 which is a constant  and the product of ${\cal F}({\cal X})\exp{ (-2\cal X)}$ is almost unity
 for $\omega$ much less than the inverse of crossing time 
 $t_0^{-1}=(r\tau_0/c)^{-1}$.  
 %In this case  
 %the harmonic pulse is not affected
 %by scattering. 
 The amplitude $B(\omega)$  decays exponentially when $\omega$ increases.
 
\section{Consequencies  of scattering effects for QPO observations:
Evidence of the resonance excitation in the QPO sources}  
 
As it is seen from  formula (4) the  amplitude of the NS 
pulsations with $\omega\sim 2\pi \times 400$ Hz  
decreases very rapidly with $\tau_0$ for 
\begin{equation}
\tau_0~~\gax ~~4[(c/R)/5\times 10^3~{\rm s}^{-1}]/[\omega/(2\pi\times 400~{\rm Hz})].
\end{equation}
%The similar result was obtained by Brainerd \& Lamb (1987) (see Eq. 5 there).
A radius $R$ of 60 km was chosen as a typical radius of
the Compton cloud derived from the observations in the frameworks of the TL
model (see TBW). 
In Figure 2 we present the rms amplitude of the high frequency QPOs as a function of
kHz frequencies for the Z-source GX 340+0 using the results of  Jonker et al. (2000). 
We compare them with the theoretical model presented by formula (4).
%Because a PDS continuum  at high frequencies are fitted by a a broad Lorentzian 
%(e.g. Wijnands \& van der Klis 1998) the resonance amplitude (unaffected by scattering)
%should be $A(\omega)\propto1/\omega$.  
The amplitude of the lower kHz QPOs is perfectly fitted by $1/\omega$ dependence (see 
the dashed line on the left
side of Fig. 2). $\chi^2_{red}=0.53$ for this fit,
but the deviation of $B(\omega)$ due to  scattering (the  solid line on the right side of Fig. 2) from the  
$1/\omega$ law (the dashed line on the right side of Fig. 2) is  clearly seen for the higher kHz QPOs.
The best fit to the data ($\chi^2_{red}=1.1$) by formula (5) with $A(\omega)\propto 1/\omega$ is obtained
for $\tau_0=2.7$ with $R/c=10^{-4}$ s. 
For $\tau_0=2.7$ the rms, $B(\omega)$ calculated  for low kHz frequencies is almost identical
to $1/\omega$ dependence ($\chi^2_{red}=0.85$).
Thus we found that  the driving force amplitude $A_{dr}$ is practically independent of frequency for
kHz QPOs. 
%\footnote{
One can argue that the driving force power for kHz QPOs  can be determined by the observed PDS at kHz  
frequencies. The PDS power law index $\beta$ typically varies  around 1
(see details in Wijnands \& van der Klis 1998). 
%If the PDS index is $-\beta)$ then the corresponding amplitude
%as a function of the QPO frequency should be $\propto \omega^{-\beta/2}$. 
The resulting resonance amplitude would be $\omega^{-(1+\beta/2)}$ with an assumption that this power 
continuum is a driving force reservoir for the resonance. As an example we fit 
this $\omega^{-(1+\beta/2)}$ model with $\beta=1$  to the data for the {\it lower} kHz QPOs 
only, where the scattering effects can be neglected (see inequality 5). 
We found that   for this model $\chi^2_{red}=1.37$. 
%This  can lead to a conclusion that the observed continuum power (PDS)
%can be the driving force reservoir for kHz QPOs. 
The rms-frequency dependence  for the lower kHz QPOs 
is not affected by scattering; it affects mainly the higher kHz QPOs (see Eq. 5).
The scattering effects make the rms-frequency dependence much steeper for the higher kHz QPOs which does not allow us
to extract the true driving force amplitude from the data (clearly, it is not feasible
to dispense with scattering and fit both QPOs solely with type of driving force model)
%his deviation is more noticeable for higher peak kHz frequencies than for 
%lower ones.
%\\
%({\bf Wei and Kent: It seems to me, in Wijnands \&
%van der Klis 1998 ApJ, 507, L63 the shape of PDS at high frequency rather flat than steep.
%WVDK wrote that the high frequency noise component is fitted by a broad Lorentzian.
%Wei,  that the intrinsic source PDS is  presumably quite flat because after resonance
%it is still has rms amplitude as $1/\omega$ })
%As is seen from formula (4), the resulting amplitude decays exponentially with increase
%of  $\omega\tau_0$ above the critical value (see inequality 5).
%In Figure 2, this behavior is shown 
%for a fixed  $\tau_0=2.7 $ which can be easily rescaled  for a variable $\tau_0$ whereas $\omega$ is fixed.  
\footnote{Fits of the rms amplitude vs frequency similar to that presented in Fig. 2 for GX 340+0, have been produced
for GX 5-1, Cyg X-2, GX 17-2 and Sco X-1.
Thus one can conclude that in all these kHz QPO observations  
the rms resonance amplitude $1/\omega$ vs frequency    is affected by the photon scattering.}.
%The shape of the resonance power vs frequency is a symmetric Lorentzian with the maximum
%at the eigenfrequency  for the linear oscillator  for high Q-values.
%In this case the observed OPO frequency as the resonance frequencies
%are identical to eigenfrequencies of the system. 
%But they rather depart from each other if there  are damping and nonlinearity in the system. 
%Titarchuk (2002) argues that  all these resonance effects are presumably seen in the RXTE observations of LMXBs.
%\subsection{The high frequency QPO limit and 
%the detection of the pulsation from the NS surface}
It is well established in the QPO observations that  there is a correlation between the QPO frequency
and the mass accretion (or the count rate). The high QPO frequency increases with the count rate
reaching a certain level and then it disappears as the low frequencies are still detected 
(Zhang et al. 1998; Mendez et al. 1999).
  This effect can be explained by the exponential decay
of the QPO amplitude with increase of $\tau_0\omega$ which is a result of the simultanenous rise of
$\tau_0$ and $\omega$ with the mass accretion rate. (see Eq. 4).   
%Furthermore  the high frequency OPOs disappear with increase of the mass accretion rate
%as the low frequencies are still detected (). This is precisely described by formula (7).
%\subsection{Chances  of }
The high frequency QPOs  presumably originate  in the outskirts 
of the Compton cloud and 
the optical depth of the QPO source location is relatively small (1-3) 
even in systems  with high accretion rate. 
In contrast,  the source of the NS pulsations 
is deeply embedded in the Compton cloud, whose  
For high mass accretion  it is hardly to expect 
the detection of the pulsations from the NS surface. The Compton cloud 
optical depth can be very
large. The NS pulsations would be totally smeared out by
the scattering  if the pulsation frequency is about 300-400 Hz and higher.  
In the next section we show that in SAX J1808.4-3658 the optical depth of spherical Compton cloud is
relatively low allowing detections.
 
 
\section{X-ray spectra of SAX J1808.4-3658 and of Other LMXBs}  
 
We revisit some  RXTE spectral data for SAX J1808.4-3658 (Gilfanov 
et al. 1998; Heindl \& Smith 1998). The spectrum (which was
constructed using data from 30411-01-06-00) is well
represented by the Comptonization model (CompTT in XSPEC, with spherical
geometry; Titarchuk 1994)
% Hua \& Titarchuk 1995) 
 along with some 
contribution from a blackbody
component and a Fe $K_{\alpha}-$ line.
%(see Figure 3).  
The best-fit parameter of the model are $\tau_0=4.5^{+0.55}_{-0.97}$, electron temperature 
$kT_e=19.5^{+7.8}_{-3.2}$  keV,  $K_{\alpha}$ energy is at $6.5\pm 0.1$ keV. 
The temperature of the seed photons for the Comptonization is within $
kT_0=0.1^{+0.4}_{-0.1}$ keV.
The temperature of the blackbody (BB) radiation (presumably from the
neutron star surface) is
$kT_{bb}=0.69\pm0.02$ keV. The quality of the fit is high with 
$\chi^2=294/322=0.91$
\footnote{Gierli\`nski, Done \& Barret (2002), hereafter GDB, also analyzed
the X-ray
spectrum of SAX J1808.4-3658 and they showed the spectrum can be fitted by
a three-component model (BB component,  Comptonization component and the 
Compton reflection).
%The color temperature of the BB component  $kT_{bb}\approx 0.65$ keV and 
%the vertical optical depth of the cylinder of
%$\tau\approx 2.7$ were
%very close to that we found. 
%They assume that the Comptonizing cloud has a shape of a cylinder with
%height-radius ratio $H/R=1$.
%Using  COMPSS Comptonization code (Poutanen \& Svensson 1996) 
%They obtained the  Comptonization parameters:
%$kT_e=43^{+9}_{-6}$ keV and the vertical optical depth of the cylinder of
%$\tau=2.7^{+0.3}_{-0.4}$.   
In general, spectral modeling is not unique  
and it is not by chance that our spectral model and  global picture are  different from that of the GDB.
Because at $kT_{bb}$ around 0.7 keV the iron is highly ionized we   conclude that the observed $K_\alpha$ 
(see \S 4 and GDB) is formed as a result of the disk reflection rather than  as  the NS  reflection.
 }.
%(we found that the reflection model is not necessary to fit to the data).
It is worth noting the relatively small optical depth $\tau=4.5$.
% and the relatively high electron temperature $kT_e=19.5$ keV. 
This can be the main reason why  the NS pulsation is detected from this source
(see inequality 5).
If  this $\tau_0$ is compared 
with that for other LMXBs sources (showing QPO features) one can find that
optical depths for a number of sources are   higher than that for  
SAX J1808.4-3658. They are 10-11 for Cyg X-2 (Kuznetsov 2002), 
Sco X-1 and  GX 340+0,  $\tau_0$ is around 6 for  4U 1728-34 (TBW). 
%Recently    Titarchuk  \& Shaposhnikov (2002) found 
%the hard state in 4U 1728-34   for which spectral parameters are similar to that 
%for SAX J1808.4-3658. Thus there is a possibility to see the NS pulsations in 
%4U 1728-34 through the Compton cloud of the optical depth a few. 
%Titarchuk \& Shaposhnikov  confirm that there is  a persistent QPO feature around 
%the burst oscillation frequency 363 Hz discovered by Strohmayer et al. (1996). 
 On average the seed photon temperature value 
 $T_0\ll T_{bb}$. Presumably the photon supply for Comptonization
mostly comes from the relatively cold disk and the blackbody emission comes 
from the NS surface. 
%The hot halo 
%is mostly concentrated inside of the sub-Keplerian transition layer.
%In other words the location of the hot electrons is not only  the accretion column
%above the  NS magnetic poles but rather the region outside of the NS surface
%associated with the disk environment. 
The Comptonization  spectra of the QPO sources where the NS pulsations are
not observed is  indirect evidence for this. %\footnote
{%We should also to point to the difference between the GDB global picture
%of the source and ours. GBD argue that the disk terminates far away from
%the NS surface because of the lack of a reflected component.
%The picture presented in this paper is quite different (Fig. 1):
%the disk (or transition region) extends down to the neutron star
%surface, while material from the disk corona is funneled onto the polar
%caps. 
%Our best-fit parameter $kT\sim 0.1$ keV suggests that the soft photon
%flux for the Comptonization comes from the disk  rather than it comes from
%the NS surface.}.
% thus we find that the our (TL) picture  is 
%also consistent with the data.}.    
van der Klis (2000) argues
that there is a striking similarity between the power spectra of black holes, atoll sources, 
Z sources and the MLP 
SAX J1808.4-3658. This would exclude any spectral formation models requiring a
material surface and would essentially imply the phenomena are generated in the
accretion disk around any low-magnetic field compact object.
But in the case of  SAX J1808.4-3658 we definitely 
see  coherent pulsations of the hard radiation
which  originates in the NS surface [Cui, Morgan \& Titarchuk (1998), hereafter CMT98] and also
we should see the downscattering effect of the hard Comptonized photons
when they  hit the relatively cold material of the NS star (see Fig.1). 

\section{Downscattering effect}
For SAX J1808.4-3658, the observed characteristics of the pulsed X-ray emission
(pulse profiles and phase lags) and the overall energy spectrum provide
useful  insight into X-ray production processes and the emission environment.
 Important information regarding the density of the NS atmosphere   
can be extracted from the time lags between different energy 
bands. The soft lags have been detected for SAX J1808.4-3658
by  CMT98\footnote{Galloway et al. 2002 also found the hard X-ray pulses arrived up to 
770 $\mu$s earlier the soft X-ray pulses in  XTE J0929-314.}.
The soft phase lag can be  due to reflection  (Compton downscattering)
of the pulsed hard radiation from the NS atmosphere.
In fact, this effect is  unavoidable when hot plasma streams converges
towards the NS magnetic poles. The hard radiation at the pole illuminates 
adjoining regions where a significant fraction of the hard radiation is 
reflected.
%  (see below and  Basko, Sunyaev \& Titarchuk 1974). 
%The  reflection efficiency (albedo) strongly depends on the 
%temperature of the NS atmosphere which can be calculated self-consistently
%(see Zel'dovich \& Shakura 1969). 
%The observed leveling-off of the fractional pulse amplitude seems to indicate
%that the intrinsic values  are approached at high energies 
%(see Fig.3 in CMT98). 
%Furthermore, the observed $K_\alpha-$iron line can be also
%produced at the NS surface as  a result of reflection.
%of X-ray radiation from the relatively cold material of the NS surface. 
%The density near the NS surface $n_e$ can be directly {\it probed} 
%by the measured time lags. 
For simplicity we assume the input hard photons
are monochromatic with energy $E_h$. The photons that emerge from 
the NS atmosphere with lower energy $E_l$ arrive at distant observer later
than those with higher energy $E_h$. {\it The delay in arrival time, $\delta t$,
is given by $u l/c$, where $u$ is the number of
scatterings undergone by  the seed hard photon with energy $E_h$} before emerging
with energy $E_l$ and $l=1/n_e\sigma_T$ is the Thomson mean free path for the
photon.
%($n_e$ is the electron density).
 %The electron temperature of the NS atmosphere as we find from the energy
%spectrum (see \S 4) is  $kT_{bb}\sim 0.7$ keV. For case where $4kT_{bb}\ll E$
The average fractional energy loss for the photon of energy $E$ 
after each scattering is nearly independent of $T_{bb}$ and it is given by
$\delta E/E\approx -(E-4kT_{bb})/m_ec^2\approx -E/m_ec^2$ (because $4kT_{bb}\ll E$), 
where $m_ec^2$ is the electron rest mass.
Integrating over multiple scatterings, we have $u=m_ec^2(1/E_l-1/E_h)$
(ST80).
The measured soft lag  scales as $E^{-1}$ as shown in Figure 3, and
levels off above 10 keV. 
In Figure 3 we also show the best fit to the data using the downscattering
model 
\begin{equation}
E^{-1}=E_h^{-1}+D\cdot\delta t.
\end{equation} 
We fit the theoretical  $E^{-1}$ as a function of $\delta t$ to the measured
quantities.
The best-fit parameters are $E_h=20$ keV and $D=1.3\times10^{-3}$ $\mu$s$^{-1}$ 
keV$^{-1}$ for which  $\chi^2_{red}=11.3/9=1.2$.
Because $D=n_ec\sigma_T/m_ec^2$ we find that {\it the NS density 
near  the surface is} $n_e=Dm_ec^2/c\sigma_T=
3.3\times10^{19}$ cm$^{-3}$.
%\footnote{The best-fit seed photon energy of 20 keV is consistent with the 
%best-fit parameters of the energy spectra which are $\tau_0=4.5$ and $kT_e=20$ keV. 
%The spectral index deduced from these
%values $\alpha=-3/2+[9/4+\pi^2/3(\tau_0+2/3)^2(511/kT_e)]^{1/2}=0.82$
%allows us to state that the maximum $E_{max}$ in $EF_{E}$ diagram is situated around $20$ keV.
%Convolved with the input photon energy distribution, the soft-lag weighted seed photon energy
%$E_h$ should be around $E_{max}$ (ST80) which is consistent with our finding using
%the soft-lag and spectral measurements.}.
The measured soft-lags in the frameworks of the downscattering models allow us to estimate 
the optical depth  $\tau$ up to what  the hard photon can penetrate illuminating
 the NS atmosphere. 
 %As we see in Figure  3 the photons with energies $\sim 20$ keV degrade to
 %the energies $\sim 5$ keV. 
It takes more than 70 scatterings for 20 keV photon to reach $~5$ keV energy band. As it is
known from the diffusion theory  (see e.g. ST80) a  photon in the course of the random
diffusion propagation to optical depth $\tau$ undergoes on average $N_{sc}\sim\tau^2$ 
scatterings. Thus using  $N_{sc}=70$ 
(the scattering number of the input photons which finally emerge) one can deduce the penetration optical depth 
$\tau_{pr}=\sqrt {N_{sc}/2}=6$.  In other words  the thickness of 
the NS reflection slab is approximately $(m_p/\sigma_T)\tau_{pr}=15$ grams cm$^{-2}$.    
Photoelectric absorption is suppressed at $kT_{bb}=0.7$ keV 
and thus one can estimate $\tau_{pr}$ neglecting the photoelectic extinction. 
%(see Chakrabarti \& Titarchuk 1995). 
The deduced $\tau_{pr}$ leads to estimate of the
reflection albedo $A=1-(3\tau_{pr}/4+1)^{-1}=0.82$ (see e.g. Sobolev 1975). 
It means that more than 80\% of the input hard photons are reflected by the NS atmosphere. 
%It is also worth noting that the gaussian lines are seen in the most of X-ray spectra 
%of LMXBs whereas the NS pulsations are not detected from these sources.
%In SAX J1808.4-3658 this line is also detected (see \S 4 and GDB).  
%But we also see these pulsations along with the
% detection of Blackbody
%component of $0.7$ keV (probably related to the NS emission) 
%and the component of 0.1 keV. 
%Because at $kT_{bb}=0.7$ keV the iron is highly
%ionized we can  conclude that the observed $K_\alpha$ 
%in SAX J1808.4-3658 
%(see \S 4 and GDB)
%is formed as a result of the disk reflection rather than  as  the NS  reflection.
 

\section{Conclusions} 
Our analysis of the timing and spectral properties of milisecond pulsar 
SAX J1808.4-3658 and comparison of them with those for LMXB QPO sources
leads us to these conclusions: 
 (1)  the detection of the NS pulsations from SAX J1808.4-3658 
 was possible because of the relatively transparent Compton cloud covering
 in this source ($\tau_0\sim 4$). For the majority of the analyzed LMXBs the Compton cloud optical depth is at
 least twice as high as that in SAX J1808.4-3658.
 The high frequency pulsations with frequencies 
 $\gax 300$ Hz are strongly wiped out by scattering in the clouds with 
 $\tau_0>4$.  SAX J1808.4-3658 is a limiting case for this detection. 
 There is a possibility of finding the NS pulsations in the
 sources that are much less luminous (than these  bright QPO LMXBs), because  in them 
 the Compton cloud is more  transparent ($\tau_0\lax 4$) for the NS pulsed
 radiation. (2) The kHz QPOs  rms vs frequency  follows the resonance law
$1/\omega$ are weakened  by  scattering in the Compton cloud. 
%The recovery of the intrinsic frequencies of the systems using the observed 
%resonance QPO frequencies does not give an accuracy better than a 
%few percents on the contrary to the previous claims.
 (3) The soft-lag measurements along with the implementation of
the downscattering model  can provide a tool  
for  density determination near  the NS 
surface.  
%We find that the NS emission region density is 
%$\sim 3\times 10^{19}$ cm$^{-3}$. The hard photons illuminated the NS 
%may  penetrate inside  the NS atmosphere to depths of 15 grams.
%It is worth noting this  atmospheric density applies near the NS polar cap where the effect of radiation
%pressure reduces the local effective gravitational acceleration by several orders of magnitude and
%increases the atmospheric scale height by that same factor.
%The exact reduction depends on details of matter and radiation flow but it is well recognized
%that, near polar caps of even low luminosity neutron stars such as Her X-1, can be locally excess of
%Eddington limit (see Arons, Klein \& Lea 1987).
%The reflection albedo of the NS atmosphere is about 80\%.

We appreciate the fruitful discussions with   Paul Ray, Sergey Kuznetsov and
 particularly, with the referee. 



\begin{thebibliography}{} 
%\bibitem[Arons et al. (1987)]{A87} 
%Arons, J., Klein, R. \& Lea, S. M.   1987,   ApJ, 312, 660  

%\bibitem[Basko et al. (1974)]{B74} 
%Basko, M.M., Sunyaev, R.A. \& Titarchuk, L. G.   1974,   A\&A, 31,249   

\bibitem[Brainerd \& Lamb (1987)]{brain}
Brainerd, C. B. \& Lamb, F.K.  1987, 317, L33
 
\bibitem[Bhattacharya \& van den Heuvel (1991)]{BV} 
Bhattacharya, D., \& van den Heuvel, E. P.J. 1991, Phys. Rep., 203,1   

%\bibitem[Bussard et al. (1988)]{Bu} 
%Bussard, R.W., Weisskopf, M.C.,  Elsner, R.F., \& Shibazaki, N.   
%1988,   ApJ, 327, 284 

\bibitem[Chandler \& Rutledge (2000)]{CR00} 
Chandler, A.M.\& Rutledge, R.E. 2000,   ApJ, 545, 1000 

%\bibitem[Chakrabarti \& Titarchuk (1995)]{CT95} 
%Chakrabarti, S.K., \& Titarchuk, L. G.   1995,   ApJ, 455, 623   
 
%\bibitem[Chandrasekhar (1960)]{chandra} 
%Chandrasekhar, S. 1960,  {\it Radiative Transfer}, New York: Dover 
 
\bibitem[ Cui et al. (2000)]{cui} 
Cui, W.,  Morgan, E.H. \& Titarchuk, L.  1998, ApJ, 504, L27 (CMT98) 

\bibitem[ Cumming et al. (2001)]{cum} 
Cumming, A.,  Zweibel, E. \& Bildsten L.  2001, ApJ, 557, 958  

\bibitem[Galloway et al. (2002)]{gall}
Galloway, D. K.  et al. 2002,  submitted to ApJ Letters, astro-ph/0206493

\bibitem[Gierlinski et al. (2002)]{GDB} 
Gierli\`nski, M., Done, C., Barret, D. 2002, MNRAS, 331, 141 (GDB)
 
\bibitem[Gilfanov  et al. (1998)]{hs} 
Gilfanov, M., Revnivtsev, M., Sunyaev, R. \& Churazov  1998, 
A\&A, 338, L83 

%\bibitem[Hua \& Titarchuk (1995)]{HT95} 
%Hua, X.M., \& Titarchuk, L. 1995, ApJ, 449, 188
 
\bibitem[ Heindl  \& Smith  (1998)]{hs} 
Heindl, M., \& Smith, D.M.  1998, ApJ, 506, L35 

%\bibitem[Jonker  et al. (2002)]{Jon02} 
%Jonker, P.G., et al. 2002, MNRAS in press, astro-ph/0202420 

\bibitem[Jonker  et al. (2000)]{Jon} 
Jonker, P.G., et al. 2000, ApJ, 537, 368 

%\bibitem[Kuznetsov \& Titarchuk (2002a)]{KT}
%Kuznetsov, S.,  \& Titarchuk, L. 2002a, ApJ, submitted (KT02)

\bibitem[Kuznetsov (2002)]{K02}
Kuznetsov, S. 2002, Astron. Lett. 28, 73 

\bibitem[Kylafis \& Klimmis (1987)]{KK}
Kylafis, N.D.,  \& Klimmis, G.S. 1987, ApJ, 323, 678

\bibitem[Landau \& Lifshitz (1971)]{LL} 
Landau, L.D. \& Lifshitz, E.M. 1971,  {\it Mechanics}, New York: Pergamon Press  

\bibitem[Markwardt et al. (2002)]{mark}
Markwardt, C. B.  et al. 2002,  submitted to ApJ Letters, astro-ph/0206491

\bibitem[Mendez et al. (1999)]{men}
Mendez, M.  et al. 1999, ApJ, 511, L46

\bibitem[Meszaros et al. (1988)]{MRB} 
Meszaros, P., Riffert, H., \& Berthiaume, G. 1988,  ApJ, 325, 204 
 
%\bibitem[Osherovich \& Titarchuk (1999)]{OT99} 
%Osherovich, V., \& Titarchuk, L. 1999, \apj, 523, L73 

\bibitem[Prudnikov et al  (1981)]{PBM} 
Prudnikov, A.P., Bruchkov, Yu.A.,\&  Marichev, O.I. 1981, 
Integrals and Series (Moscow: Nauka)  


%\bibitem[Shakura \& Sunyaev  (1973)]{ss73} 
%Shakura, N.I.,  \& Sunyaev, R.A. 1973, A\&A, 24, 337 

\bibitem[Sobolev (1975)]{sob}
Sobolev, V.V. 1975, Light scattering in Atmospheres (Oxford: Pergamon)

%\bibitem[Strohmayer et al.  (1997)]{S97} 
%Strohmayer, T.E., Jahoda, K., Giles, B., \& Lee, U.  1997, ApJ, 486, 355 

\bibitem[Strohmayer et al.  (1996)]{S96} 
Strohmayer, T.E., et al.  1996, ApJ, 469, L9 

%\bibitem[Strohmayer et al.  (1996)]{S96} 
%Strohmayer, T.E., Jahoda, K., Giles, B., \& Lee, U.  1996b, IAU Circ. 6494 
 
\bibitem[Sunyaev \& Titarchuk (1980)]{sun80} Sunyaev, R. \& 
Titarchuk, L., 1980, A\&A, 86, 121 (ST80) 

%\bibitem[Titarchuk \& Shaposhnikov (2002)]{TS02} 
%Titarchuk, L. \& Shaposhnikov, N. 2002, in preparation  

%\bibitem[Titarchuk (2002)]{T02} 
%Titarchuk, L. 2002, \apj, submitted  
  
\bibitem[Titarchuk, et al  (2001)]{TBW} 
Titarchuk, L. G., Bradshaw, C.F.,  \& Wood, K.S. 2001, ApJ, 560,  L55 (TBW)

%\bibitem[Titarchuk, \& Osherovich (2001)]{TO01} 
%Titarchuk, L., \& Osherovich, V. 2001, \apj, 555 , L55 

\bibitem[Titarchuk, et al  (2001)]{Tetal} 
Titarchuk, L. G., Bradshaw, C.F.,  Geldzahler, B.J., \& Fomalont, E.B. 
2001, ApJ, 555,  L45 

%\bibitem[Titarchuk, \& Osherovich (2000)]{TO00} 
%Titarchuk, L., \& Osherovich, V. 2000, \apj,  537, L39 

%\bibitem[Titarchuk, \& Osherovich (1999)]{TO99} 
%Titarchuk, L., \& Osherovich, V. 1999, \apj,  518, L95 

%\bibitem[Titarchuk, Osherovich, \& Kuznetsov (1999)]{TO99} 
%Titarchuk, L., Osherovich, V., \& Kuznetsov, S. 1999, \apj,  525, L129 (TOK)

 \bibitem[Titarchuk, Lapidus \& Muslimov (1998)]{TLM98} 
Titarchuk, L., Lapidus, I.,\& Muslimov, A. 1998, \apj,  499, 315 
(TLM98)
 
\bibitem[Titarchuk, (1994)]{T94} 
Titarchuk, L. 1994, ApJ, 434, 570 
 
\bibitem[Vaughan, et al. (1994)]{V94} 
Vaughan, B. A., et al. 1994, ApJ, 435, 362 
 
\bibitem[van der Klis (2000)]{vdk}
van der Klis, M. 2000, ARA\&A, 38, 717

%\bibitem[Wang \& Schlickeiser (1987)]{w87} 
%Wang, Y.-M., \& Schlickeiser, R.  1987, ApJ, 313, 200 

\bibitem[ Wijnands \& van der Klis (1998)]{wv}
Wijnands, R. \& van der Klis, M. 1998, ApJ, 507, L63

\bibitem[Wood et al. (2001)]{w01} 
Wood, K.S., et al.  2001, ApJ, 563, 246 

\bibitem[Wood et al. (1991)]{w91} 
Wood, K.S., et al.  1991, ApJ, 379, 295 
 
\bibitem[Wood et al. (1988)]{w88} 
Wood, K.S., Ftaclas, C., \& Kearney, M.  1988, ApJ, 324, L63 

\bibitem[Zhang et al. (1998)]{zh}
Zhang, W., Smale, A. P., Strohmayer, T.E., \& Swank, J.H. 1998,  ApJ, 500, L171 

%\bibitem[Zel'dovich \& Shakura (1969)]{ZS}
%Zel'dovich, Ya.B., \& Shakura, N.I. 1969, AZh, 46, 225 (English transl. in Soviet Astron.
%13, 175)
\end{thebibliography} 
 
%\appendix
%\section{Analytical Calculation of Integral $I$ }
%In fact, the integral $I$  can be reduced to the integral
% $I_{\alpha}=\int_0^{\infty}u^{-\alpha-1/2}\exp{(-pu-q/u)}du$ which
%  is calculated exactly for integer $\alpha=n$.

%For n=0,  $I_0=2p^{-1/2}\exp{[-2(pq)^{1/2}]}\int_0^{\infty}\exp{(-z^2)}dz 
%- (q/p)^{1/2}I_1$ where $z=p^{1/2}u^{1/2}-q^{1/2}u^{-1/2}$.  
%Keeping in mind that the first integral is $\sqrt\pi$ (see a definition of $z$
%above) and $I_1=-\partial I_0/\partial q$  we come to the 
%differential equation for $I_0$
%\begin{equation}
%\sqrt{\frac{q}{p}}\frac{\partial I_0}{\partial q}-I_0=-2
%\sqrt{\frac{\pi}{p}}
%\exp{(-2\sqrt{pq})}
%\end{equation}
%for which a particular solution is $I_0=(\pi/p)^{1/2}\exp[-2 (pq)^{1/2}]$.
%For any n, $I_n$ can be calculated using derivatives over q, namely
%$I_1=-\partial I_0/\partial q$, $I_2=-\partial I_1/\partial q$ and so on. 
%For any non-integer $\alpha$, $I_{\alpha}$ is expressed through the modified
%Bessel functions  $K_{\alpha}(2\sqrt{pq})$ 
%(see also Prudnikov, Bruchkov \& Marichev 1981, formulas 2.3.16).
%$K_{\alpha}(2\sqrt{pq})$  has an exponential behavior $\exp[-2 (pq)^{1/2}]$
%for  $2 (pq)^{1/2}>1$, which is precisely our case (see formula 7 for 
%$\cal X$).
\clearpage 
 
 
\begin{figure} 
\epsscale{0.7} 
\plotone{f1.eps} 
\caption{Schematic picture of the model geometry SAX J1808.4-3658.
The hot flow forming at the outer boundary of the transition layer streams
towards neutron star along magnetic field lines. The reflection of the hard
radiation of the flow from the neutron star surface results in soft time lags
of the pulsed emission.} 
\label{fig1} 
\end{figure} 

\begin{figure} 
\epsscale{0.7} 
\plotone{f2.eps} 
\caption{ Examples of RMS amplitude vs QPO frequency for lower and higher kHz peaks
for GX 340+0 (Jonker et al. 2000). Two solid lines (for lower and higher peaks respectively) 
are theoretical curves for which  scattering effects  are taken into account. Dashed lines are for $1/\nu$ law 
and dash-dotted  line is  for  $\nu^{-3/2}$  law.
}
\label{fig2} 
\end{figure} 

%\begin{figure}
%\includegraphics[width=5in,angle=-90]{f3.eps}
%\caption{\label{spectrum}The Best-fit spectral model for SAX J1808.4-3658. Also shown are
%individual components of the model: Comptonization (CompTT in XSPEC), 
%blackbody, and Gaussian function
%}
%\end{figure}
 
%\begin{figure} 
%\epsscale{0.7} 
%\plotone{f3.eps} 
%\caption{The Best-fit spectral model for SAX J1808.4-3658. Also shown are
%individual components of the model: Comptonization (CompTT in XSPEC), 
%blackbody, and Gaussian function.}
%\label{fig3} 
%\end{figure} 
 
\begin{figure} 
\epsscale{0.7} 
\plotone{f3.eps} 
\caption{Measured soft X-ray lags with respect to 20-30 keV band along with 
the best-fit downscattering model.} 
\label{fig3} 
\end{figure} 
 
  
\clearpage 
 
\end{document} 


astro-ph0207552.extracted_bib
\begin{thebibliography}{} 
%\bibitem[Arons et al. (1987)]{A87} 
%Arons, J., Klein, R. \& Lea, S. M.   1987,   ApJ, 312, 660  

%\bibitem[Basko et al. (1974)]{B74} 
%Basko, M.M., Sunyaev, R.A. \& Titarchuk, L. G.   1974,   A\&A, 31,249   

\bibitem[Brainerd \& Lamb (1987)]{brain}
Brainerd, C. B. \& Lamb, F.K.  1987, 317, L33
 
\bibitem[Bhattacharya \& van den Heuvel (1991)]{BV} 
Bhattacharya, D., \& van den Heuvel, E. P.J. 1991, Phys. Rep., 203,1   

%\bibitem[Bussard et al. (1988)]{Bu} 
%Bussard, R.W., Weisskopf, M.C.,  Elsner, R.F., \& Shibazaki, N.   
%1988,   ApJ, 327, 284 

\bibitem[Chandler \& Rutledge (2000)]{CR00} 
Chandler, A.M.\& Rutledge, R.E. 2000,   ApJ, 545, 1000 

%\bibitem[Chakrabarti \& Titarchuk (1995)]{CT95} 
%Chakrabarti, S.K., \& Titarchuk, L. G.   1995,   ApJ, 455, 623   
 
%\bibitem[Chandrasekhar (1960)]{chandra} 
%Chandrasekhar, S. 1960,  {\it Radiative Transfer}, New York: Dover 
 
\bibitem[ Cui et al. (2000)]{cui} 
Cui, W.,  Morgan, E.H. \& Titarchuk, L.  1998, ApJ, 504, L27 (CMT98) 

\bibitem[ Cumming et al. (2001)]{cum} 
Cumming, A.,  Zweibel, E. \& Bildsten L.  2001, ApJ, 557, 958  

\bibitem[Galloway et al. (2002)]{gall}
Galloway, D. K.  et al. 2002,  submitted to ApJ Letters, astro-ph/0206493

\bibitem[Gierlinski et al. (2002)]{GDB} 
Gierli\`nski, M., Done, C., Barret, D. 2002, MNRAS, 331, 141 (GDB)
 
\bibitem[Gilfanov  et al. (1998)]{hs} 
Gilfanov, M., Revnivtsev, M., Sunyaev, R. \& Churazov  1998, 
A\&A, 338, L83 

%\bibitem[Hua \& Titarchuk (1995)]{HT95} 
%Hua, X.M., \& Titarchuk, L. 1995, ApJ, 449, 188
 
\bibitem[ Heindl  \& Smith  (1998)]{hs} 
Heindl, M., \& Smith, D.M.  1998, ApJ, 506, L35 

%\bibitem[Jonker  et al. (2002)]{Jon02} 
%Jonker, P.G., et al. 2002, MNRAS in press, astro-ph/0202420 

\bibitem[Jonker  et al. (2000)]{Jon} 
Jonker, P.G., et al. 2000, ApJ, 537, 368 

%\bibitem[Kuznetsov \& Titarchuk (2002a)]{KT}
%Kuznetsov, S.,  \& Titarchuk, L. 2002a, ApJ, submitted (KT02)

\bibitem[Kuznetsov (2002)]{K02}
Kuznetsov, S. 2002, Astron. Lett. 28, 73 

\bibitem[Kylafis \& Klimmis (1987)]{KK}
Kylafis, N.D.,  \& Klimmis, G.S. 1987, ApJ, 323, 678

\bibitem[Landau \& Lifshitz (1971)]{LL} 
Landau, L.D. \& Lifshitz, E.M. 1971,  {\it Mechanics}, New York: Pergamon Press  

\bibitem[Markwardt et al. (2002)]{mark}
Markwardt, C. B.  et al. 2002,  submitted to ApJ Letters, astro-ph/0206491

\bibitem[Mendez et al. (1999)]{men}
Mendez, M.  et al. 1999, ApJ, 511, L46

\bibitem[Meszaros et al. (1988)]{MRB} 
Meszaros, P., Riffert, H., \& Berthiaume, G. 1988,  ApJ, 325, 204 
 
%\bibitem[Osherovich \& Titarchuk (1999)]{OT99} 
%Osherovich, V., \& Titarchuk, L. 1999, \apj, 523, L73 

\bibitem[Prudnikov et al  (1981)]{PBM} 
Prudnikov, A.P., Bruchkov, Yu.A.,\&  Marichev, O.I. 1981, 
Integrals and Series (Moscow: Nauka)  


%\bibitem[Shakura \& Sunyaev  (1973)]{ss73} 
%Shakura, N.I.,  \& Sunyaev, R.A. 1973, A\&A, 24, 337 

\bibitem[Sobolev (1975)]{sob}
Sobolev, V.V. 1975, Light scattering in Atmospheres (Oxford: Pergamon)

%\bibitem[Strohmayer et al.  (1997)]{S97} 
%Strohmayer, T.E., Jahoda, K., Giles, B., \& Lee, U.  1997, ApJ, 486, 355 

\bibitem[Strohmayer et al.  (1996)]{S96} 
Strohmayer, T.E., et al.  1996, ApJ, 469, L9 

%\bibitem[Strohmayer et al.  (1996)]{S96} 
%Strohmayer, T.E., Jahoda, K., Giles, B., \& Lee, U.  1996b, IAU Circ. 6494 
 
\bibitem[Sunyaev \& Titarchuk (1980)]{sun80} Sunyaev, R. \& 
Titarchuk, L., 1980, A\&A, 86, 121 (ST80) 

%\bibitem[Titarchuk \& Shaposhnikov (2002)]{TS02} 
%Titarchuk, L. \& Shaposhnikov, N. 2002, in preparation  

%\bibitem[Titarchuk (2002)]{T02} 
%Titarchuk, L. 2002, \apj, submitted  
  
\bibitem[Titarchuk, et al  (2001)]{TBW} 
Titarchuk, L. G., Bradshaw, C.F.,  \& Wood, K.S. 2001, ApJ, 560,  L55 (TBW)

%\bibitem[Titarchuk, \& Osherovich (2001)]{TO01} 
%Titarchuk, L., \& Osherovich, V. 2001, \apj, 555 , L55 

\bibitem[Titarchuk, et al  (2001)]{Tetal} 
Titarchuk, L. G., Bradshaw, C.F.,  Geldzahler, B.J., \& Fomalont, E.B. 
2001, ApJ, 555,  L45 

%\bibitem[Titarchuk, \& Osherovich (2000)]{TO00} 
%Titarchuk, L., \& Osherovich, V. 2000, \apj,  537, L39 

%\bibitem[Titarchuk, \& Osherovich (1999)]{TO99} 
%Titarchuk, L., \& Osherovich, V. 1999, \apj,  518, L95 

%\bibitem[Titarchuk, Osherovich, \& Kuznetsov (1999)]{TO99} 
%Titarchuk, L., Osherovich, V., \& Kuznetsov, S. 1999, \apj,  525, L129 (TOK)

 \bibitem[Titarchuk, Lapidus \& Muslimov (1998)]{TLM98} 
Titarchuk, L., Lapidus, I.,\& Muslimov, A. 1998, \apj,  499, 315 
(TLM98)
 
\bibitem[Titarchuk, (1994)]{T94} 
Titarchuk, L. 1994, ApJ, 434, 570 
 
\bibitem[Vaughan, et al. (1994)]{V94} 
Vaughan, B. A., et al. 1994, ApJ, 435, 362 
 
\bibitem[van der Klis (2000)]{vdk}
van der Klis, M. 2000, ARA\&A, 38, 717

%\bibitem[Wang \& Schlickeiser (1987)]{w87} 
%Wang, Y.-M., \& Schlickeiser, R.  1987, ApJ, 313, 200 

\bibitem[ Wijnands \& van der Klis (1998)]{wv}
Wijnands, R. \& van der Klis, M. 1998, ApJ, 507, L63

\bibitem[Wood et al. (2001)]{w01} 
Wood, K.S., et al.  2001, ApJ, 563, 246 

\bibitem[Wood et al. (1991)]{w91} 
Wood, K.S., et al.  1991, ApJ, 379, 295 
 
\bibitem[Wood et al. (1988)]{w88} 
Wood, K.S., Ftaclas, C., \& Kearney, M.  1988, ApJ, 324, L63 

\bibitem[Zhang et al. (1998)]{zh}
Zhang, W., Smale, A. P., Strohmayer, T.E., \& Swank, J.H. 1998,  ApJ, 500, L171 

%\bibitem[Zel'dovich \& Shakura (1969)]{ZS}
%Zel'dovich, Ya.B., \& Shakura, N.I. 1969, AZh, 46, 225 (English transl. in Soviet Astron.
%13, 175)
\end{thebibliography}