A model for the waveform behavior of accreting millisecond pulsars: Nearly aligned magnetic fields and moving emission regions

TL;DR: In this article, the authors investigated a model of the accreting millisecond X-ray pulsars and showed that the magnetic poles of these stars are close to their spin axes, and the gas is channeled along different field lines to different locations on the stellar surface.

Abstract: We investigate further a model of the accreting millisecond X-ray pulsars we proposed earlier. In this model, the X-ray-emitting regions of these pulsars are near their spin axes but move. This is to be expected if the magnetic poles of these stars are close to their spin axes, so that accreting gas is channeled there. As the accretion rate and the structure of the inner disk vary, gas is channeled along different field lines to different locations on the stellar surface, causing the X-ray-emitting areas to move. We show that this "nearly aligned moving spot model" can explain many properties of the accreting millisecond X-ray pulsars, including their generally low oscillation amplitudes and nearly sinusoidal waveforms; the variability of their pulse amplitudes, shapes, and phases; the correlations in this variability; and the similarity of the accretion- and nuclear-powered pulse shapes and phases in some. It may also explain why accretion-powered millisecond pulsars are difficult to detect, why some are intermittent, and why all detected so far are transients. This model can be tested by comparing with observations the waveform changes it predicts, including the changes with accretion rate.

Summary

1. INTRODUCTION

• Highly periodic millisecond X-ray oscillations have been detected with high confidence in 22 accreting neutron stars in low-mass X-ray binary systems , using the Rossi Xray Timing Explorer (RXTE) satellite (see Lamb & Boutloukos 2008).
• Accretion-powered millisecond oscillations have so far been detected in 10 AMXPs.
• Emission from a spot close to the spin axis has only a small asymmetry and therefore produces only weak modulation.
• These effects may explain the fact that accretion-powered X-ray oscillations have not yet been detected in many accreting neutron stars that are thought to have millisecond spin periods and dynamically important magnetic fields.
• If the pulse amplitude and phase variations observed in AMXPs are caused by motion of the emitting area, they should be correlated.

2.1. Modeling the X-ray Emission

• In the radiating spot model of AMXP X-ray emission, the waveform seen by a distant observer depends on the sizes, shapes, and positions of the emitting regions on the stellar surface; the beaming pattern of the radiation; the compactness, radius, and spin rate of the star; and the direction from which the star is observed.
• If some of the accreting plasma were to become hot, the forces exerted on it by the stellar magnetic field would tend to drive it toward the star’s magnetic equator (Michel 1977), causing it to impact the stellar surface in an annulus around the star’s magnetic equator.
• The authors find that in many cases these waveforms can be approximated by the waveforms generated by a circular, uniformly emitting spot located at the centroid of the emitting region.
• An observer may see radiation from a single spot either because the accretion flow pattern strongly favors one pole of a dipolar stellar magnetic field over the other, or because the observer’s view of one pole is blocked by the inner disk or by accreting plasma in the star’s magnetosphere (see McCray & Lamb 1976; Basko & Sunyaev 1976).

2.2. Computing X-ray Waveforms

• The X-ray waveforms calculated here assume that radiation propagating from emitting areas on the stellar surface reaches the observer without interacting with any intervening matter.
• The authors describe the emission from the stellar surface using coordinates centered on the star.
• When considering emission from two spots, the authors somewhat arbitrarily identify one as the primary spot and the other as the secondary spot.
• The authors carried out many calculations to test and verify the computer code used to obtain the results they report here.
• The numerical results agreed with the analytical results.

2.3. Constructing Pulse Profiles

• The X-ray flux seen by a given observer will evolve continuously in time as the star rotates and the emission from the stellar surface changes, generating the observed waveform W (t).
• As noted in Section 2.1, the accretion flow from the inner disk to the stellar surface is expected to vary on timescales at least as short as the ∼1 ms dynamical timescale near the neutron star, which will cause the sizes, shapes, and positions of the emitting regions, and therefore the observed waveform, to vary on these timescales.
• If the data are folded with a period Pf that is chosen to agree as closely as possible with the local, approximate repetition period P (ti) of the waveform, one can construct a time sequence of pulse profiles WP (φ, ti); here φ is the pulse phase over one cycle.
• The waveform seen by an observer in the star’s rotation equator viewing two identical antipodal spots in the rotation equator should be the same at 180◦ as at 0◦, but this is not the case for their waveform for this geometry .
• This is not quite true for the waveforms reported by Pechenick et al. (1983).

3. OSCILLATION AMPLITUDES

• As discussed in Section 1, the fractional amplitudes of the accretion-powered oscillations of most AMXPs are typically ∼1%–2%, but the amplitudes of several AMXPs are sometimes as large as ∼10%–20%.
• A successful model of the accretionpowered oscillations of the AMXPs should therefore be able to explain oscillation amplitudes as low as ∼1%–2% without requiring a special viewing angle or stellar structure and should also be able to explain the higher amplitudes sometimes seen.
• The second harmonic of the fundamental oscillation frequency has been detected in seven of the 10 known AMXPs, but it is typically 10 times weaker than the fundamental, although in a few cases it is not this weak and in one case, SAX J1808.4−3658, it is sometimes stronger than the fundamental.
• Measured oscillation amplitudes are likely to be smaller than those shown in the figures in this section, which are for stable spots fixed on the stellar surface.
• Such rapid pulse shape variations will appear as increased background noise, reducing the apparent amplitude of the oscillations (see Lamb et al. 1985).

3.1. Dependence on Spot Inclination

• The precise distance emitting regions can be from the spin axis and still produce oscillation amplitudes as low as the ∼1%–2% amplitudes often observed in the AMXPs depends on the beaming pattern of the emission.
• The authors first discuss the waveforms produced by isotropic emission and then the waveforms produced by other beaming patterns.
• The amplitude of the second harmonic is 2% for all observers only if the spots are within 15◦ of the spin axis.

3.2. Dependence on Spot Size

• Larger emitting spots tend to produce lower oscillation amplitudes than smaller spots, but even very large spots must be centered close to the stellar spin axis in order to explain the low oscillation amplitudes observed in the AMXPs, unless almost the entire stellar surface is uniformly emitting.
• All the curves in this figure are for their reference star, viewed at an inclination of 45◦.
• The amplitude of the oscillation produced by a single spot inclined 45◦ from the spin axis decreases by only ∼10% as the spot radius increases from 5◦ to 45◦.
• Two antipodal spots on their reference star can produce a fractional modulation as low as ∼2% for most observing directions only if they have radii 75◦, which means that all of the stellar surface is uniformly emitting except for a band around the star with a total width 30◦.
• Another difficulty with attributing the low fractional amplitudes often observed to large emitting areas is that a substantial number of AMXPs that are observed to have fractional amplitudes ∼1%– 2% at some times are observed to have much larger fractional amplitudes ∼15%–20% at other times.

3.3. Dependence on Stellar Compactness

• The fractional modulation produced by a given emission pattern is generally smaller for more a compact star (see, e.g. Pechenick et al. 1983; Strohmayer 1992).
• Strong gravitational focusing can increase the fractional modulation seen by some observers if the star is very compact (R 3.5 M).
• Nor can the failure so far to detect accretionpowered oscillations in some nuclear-powered AMXPs be explained unless the accretion-powered emission comes from areas very close to the stellar spin axis.
• A further difficulty in attributing the generally low fractional modulations of the AMXPs to high stellar compactness is that several of the AMXPs that exhibit fractional modulations ∼1%– 2% at some times exhibit fractional modulations ∼15%–25% only a few hours or days later.

3.4. Dependence on Stellar Spin Rate

• Most of the results discussed in this section are for stars spinning at 400 Hz.
• Other things being equal, stars spinning more rapidly will produce oscillations with larger fractional amplitudes, because their higher surface velocities will produce larger Doppler boosts and greater aberration, making their radiation patterns more asymmetric (Miller & Lamb 1998; Braje et al. 2000).
• Conversely, stars spinning more slowly tend to produce oscillations with smaller fractional amplitudes.
• The dependence of the fractional amplitude on the stellar spin rate is weak.
• The basic conclusions reached in this section are valid for the full range of AMXP spin rates observed.

4. PULSE AMPLITUDE AND PHASE VARIATIONS

• If interpreted as caused by changes in the stellar spin rate, the observed phase variations would imply frequency variations more than a factor of 10 larger than expected for the largest accretion torques and the smallest inertial moments thought possible for these stars.
• In several AMXPs, the phase variations are correlated with the amplitude variations, for some amplitude ranges.
• A successful model of the accretion-powered oscillations of the AMXPs should provide a consistent explanation of these properties of the oscillations.
• Changes in the longitude (stellar azimuth) of the emitting area shift the phases of the harmonic components of the pulse but do not affect their amplitudes.
• The authors show further that if the emitting area is close to the spin axis and moves in the azimuthal direction by even a small distance, the phases of the Fourier components of the pulse will shift by a large amount.

4.1. Pulse Amplitude Variations

• Figure 4 shows the total fractional rms amplitude and the fractional rms amplitude of the second harmonic (first overtone) of the spin frequency for pulses produced by isotropic emission from a single stable spot and from two stable antipodal spots, as functions of the inclination of the primary spot relative to the spin axis.
• Whether the observer sees emission from a single area or from two antipodal areas, the total fractional amplitude of the oscillation will increase if the inclination of an emitting region initially at a low inclination increases.
• The accretion-powered oscillations observed in XTE J1814−338 and XTE J1807−294 occasionally have fractional amplitudes as large as 11% and 19%, respectively, although they are usually much smaller (Chung et al.
• The harmonic amplitudes of AMXP pulses are expected to vary on these timescales.

4.2. Pulse Phase Variations

• A change in the longitude (stellar azimuth) of the emitting region alters the arrival time of the pulse.
• This shifts the measured phases of all the Fourier components by the same amount, relative to their phases if the pulse, were produced by an emitting area fixed on the surface of a star rotating at a constant rate.
• As the emitting spot moves around the path shown in Figure 5, its inclination increases from 2◦ to 22◦.
• As described above, the change in the inclination of the spot as it moves around the path shown in Figure 5 also contributes to the phase shifts.
• The phase residuals of the first and second harmonic components of the pulses of XTE J1814−338 appear to be anticorrelated with its X-ray flux (Chung et al. 2008; Papitto et al. 2007), making it a good candidate for this kind of study.

4.3. Correlated Pulse Amplitude and Phase Variations

• If the emitting areas of the AMXPs are near their spin axes and move around with time, the amplitudes and phases of their pulses should show two types of correlated behavior.
• In the example shown, the phase residuals vary by ∼0.3 cycles when the fractional amplitude is ∼0.02 but by only ∼0.03 cycles when the fractional amplitude is ∼0.1.
• A second expectation in the moving spot model discussed here is that the pulse arrival time or phase residuals are likely to form a track in the phase-residual versus pulse-amplitude plane, especially if the change in the pulse amplitude is large.
• The path on the stellar surface along which the emitting area moves as the accretion rate changes depends on the details of the accretion flow from the inner disk to the stellar surface that cannot yet be determined from first principles.
• If instead the phase residuals of different Fourier components of the pulse evolve very differently with increasing pulse amplitude, this is an indication that the emitting area is moving along a path that produces very little change in the area’s longitude and/or a substantial change in its shape or the radiation-beaming pattern.

4.4. Accretion- and Nuclear-powered Oscillations

• The close agreement of the pulse profiles and phases of the accretion- and nuclear-powered (X-ray burst) oscillations observed in SAX J1808.4−3658 (Chakrabarty et al. 2003) and XTE J1814−338 (Strohmayer et al. 2003) strongly suggests that in these stars, both types of oscillation are produced by X-ray emission from nearly the same area on the stellar surface.
• If this is so, it implies that in these AMXPs thermonuclear burning is concentrated near the magnetic poles onto which accreting matter is falling.
• It also implies that long-term variations in the phase residuals of the two types of oscillations should track one another in these pulsars and should also be correlated with variations in the X-ray flux and spectrum, because both types of variations are produced by changes in the accretion flow through the inner disk.
• If this interpretation proves correct, the locations and movements of the emitting areas can be determined from the observed phase and amplitude variations.
• In Section 5.1, the authors emphasize that a mechanism that drives a star’s magnetic poles toward its spin axis can greatly reduce the dipole component of the star’s magnetic field without reducing significantly its strength.

4.5. Effects of Rapid Spot Movements

• The authors results show that motion of the emitting area on the stellar surface generally changes both the amplitudes and the phases of the Fourier components of the pulse profile.
• As noted in Section 2.1, the position of the emitting area is expected to reflect the accretion rate and structure of the inner disk, and is therefore expected to change on timescales at least as short as the ∼0.1 ms dynamical time near the neutron star.
• Here the authors discuss the expected effects of fluctuations in the position of the emitting area on timescales shorter than the time required to construct a pulse profile.
• These pulse phase and amplitude fluctuations are likely to be greater when the emitting area is near the spin axis, because there a displacement by a given distance produces a larger change in the pulse phase and, for many geometries, in the pulse amplitude.
• This noise can in principle be detected, especially because its strength is expected to be anticorrelated with the pulse amplitude and to depend in a systematic way on the X-ray flux and spectrum of the pulsar.

4.6. Undetected and Intermittent Pulsations

• The results presented in Section 3 show that if the emitting area is very close to the spin axis and remains there, the amplitude of X-ray oscillations at the stellar spin frequency or its overtones may be so low that they are undetectable.
• In addition, rapid variations in the shape and phase of pulses are expected to be stronger when the emitting area is very close to the spin axis.
• The noise produced by these fluctuations may— in combination with other effects, such as reduction of the modulation fraction by scattering in circumstellar gas—further reduce the detectability of accretion-powered oscillations in neutron stars with millisecond spin periods (Lamb et al. 1985; Miller 2000).
• A change in the accretion flow within the magnetosphere can suddenly channel gas to the stellar surface farther from the spin axis, causing the centroid of the emitting area to move away from the axis.
• This will increase the amplitude of the oscillation, potentially making a previously undetectable oscillation detectable.

5. DISCUSSION

• The results presented in previous sections show that a model of AMXPs in which the X-ray-emitting areas are close to the spin axis and move around on the stellar surface can explain many of their properties.
• Emitting areas close to the spin axis are to be expected if the magnetic poles of the AMXPs are close to their spin axes, causing accreting gas to be channeled there.
• The authors first discuss mechanisms that may cause the magnetic poles of AMXPs to be close to their spin axes.
• The authors then point out that this picture of magnetic field evolution may also explain why AMXPs in which accretion-powered oscillations have been detected are transient X-ray sources.
• The authors also discuss several observational tests of this model.

5.1. Movement of Magnetic Poles Toward the Spin Axis

• The neutron vortices in the fluid core of a spinning neutron star are expected to move radially inward if the star is spun up.
• If the star’s north and south magnetic poles are in opposite rotation hemispheres when spin-up begins, the inward motion of vortices will drag them toward opposite spin poles.
• The magnetic flux threading the fluid core is conserved as the magnetic field is squeezed.
• The full strength M2 of the surface magnetic field will therefore be ≈ (R/a)2 ≈ 103 times larger than the strength M2 of its component.
• Another consequence of this picture of magnetic field evolution is that both accretion-powered (X-ray) and rotationpowered millisecond pulsars may have surface magnetic fields ∼1011–1012 G.

5.2. Why AMXPs are Transients

• The picture of the X-ray emission and magnetic field evolution of AMXPs that the authors have outlined here suggests a possible explanation for why all AMXPs found so far are transients.
• These systems have very low long-term average mass transfer rates, but binary modeling suggests that the mass transfer rates were higher in the past (see, e.g., Bildsten & Chakrabarty 2001 for a discussion in the context of SAX J1808.4−3658).
• If stars such as these are initially spun up to high spin rates, so that their magnetic poles are forced very close to their spin axes, they would appear similar to the accreting neutron stars in low-mass X-ray binary systems in which accretion-powered oscillations have not been detected.
• When their accretion rates later decrease, and magnetic dipole and other braking torques cause them to spin down, their magnetic poles will be forced away from the rotation axis, and their accretion-powered oscillations will become detectable.

5.3. Comparison with the Properties of MRPs

• As explained above, either orientation is consistent with their magnetic poles being driven close to their spin axes by neutron vortex motion during spin-up as AMXPs.
• Analysis and modeling of the waveforms of the thermal Xray emission from MRPs may provide better constraints on their magnetic field geometries.
• Recent modeling of the high (30% to 50% rms) amplitude X-ray oscillations observed in three nearby MRPs using Comptonized emission and two antipodal or nearly antipodal hot spots is consistent with their emitting regions being far from their spin axes (Bogdanov et al. 2007, 2008).
• The authors note that all three pulsars have relatively low (∼150 Hz–200 Hz) spin rates and may therefore have been spun down by a factor ∼3 from their maximum spin frequencies after spin-up.

5.4. Other Observational Tests

• In the nearly aligned moving spot model of AMXP X-ray emission, the properties of X-ray pulses (e.g., their amplitudes, harmonic content, and arrival times) should be functions of the pulsar’s X-ray luminosity and spectrum.
• The location of the emitting area on its favored path will depend on the accretion flow through the inner disk and will in turn determine the position of pulses in the phase-residual versus pulse-amplitude plane.
• If this is the case, the model predicts that longer-term (days–weeks) variations of the phase residuals of the accretion- and nuclearpowered oscillations should be correlated with one another and with longer term variations of the pulsar’s X-ray luminosity and spectrum, because all three variations are expected to be related to changes in the accretion flow through the inner disk.
• In the moving spot model, the motion of the emitting area that produces this excess noise also affects the amplitudes, harmonic content, and arrival times of the X-ray pulses, and is related to the X-ray luminosity and spectrum.
• If AMXPs do have surface magnetic fields as strong as ∼1011–1012 G, as suggested in Section 5.1, their keV spectra may show strong-magnetic-field features.

6. CONCLUSIONS

• In previous sections, the authors have explored in some detail the nearly aligned moving spot model of AMXP X-ray emission.
• Variability of pulse amplitudes, shapes, and arrival times.
• The authors computations show that motion of the emitting area on the stellar surface on timescales longer than the spin period usually changes the amplitudes and the phases of the harmonic components of the theoretical pulse profile.
• In Section 3.1, the authors showed that if the emitting areas of some AMXPs are very close to the spin axis and remain there, the amplitudes of the oscillations they would produce can be ∼0.5% or less, making them undetectable with current instruments.
• Rapid X-ray flux variations will make accretion-powered oscillations at the spin frequency more difficult to detect.

Figures

Figure 1. Top panels (left to right): contour plots of the fractional rms amplitudes of the first harmonic (fundamental) and second harmonic (first overtone) components of the pulse profiles and the total fractional rms amplitude produced by a single stable spot, as a function of the observer’s inclination and the inclination of the primary spot relative to the spin axis. Bottom panels (left to right): corresponding contour plots for the pulse profiles produced by two stable antipodal spots. All plots assume spot radii of 25◦ and a 1.4M star with a radius of 5M spinning at 400 Hz. The heavier lines highlight the contours where the fractional flux variation is 2%. This figure shows that emitting regions on or near the stellar surface can produce oscillation amplitudes as low as ∼1%–2% for a substantial range of viewing directions only if they are located within a few degrees of the stellar spin axis.

Figure 4. Fractional rms amplitudes of the full bolometric pulse profile (left) and the second harmonic (first overtone) component of the pulse profile (right) produced by isotropic emission from a single stable spot and from two stable antipodal spots, as functions of the inclination of the primary spot relative to the spin axis, for spot radii of 25◦, a 1.4M star with a radius of 5M spinning at 400 Hz, and the indicated observer inclinations i. This figure shows that if the inclination of the emitting region is small, modest changes in its inclination can produce large changes in the observed amplitude of the oscillation.

Figure 2. Contour plots of the total fractional rms amplitude produced by isotropic emission from a single stable spot (left-hand panel) and from two stable antipodal spots (right-hand panel), as a function of the spot radius and the inclination of the primary spot relative to the spin axis, for a 1.4M star with a radius of 5M spinning at 400 Hz observed at an inclination of 45◦. The heavier lines highlight the contours where the total flux modulation is 2%. This figure demonstrates that even very large spots must be centered close to the stellar spin axis in order to explain the low oscillation amplitudes observed in the AMXPs, unless almost the entire stellar surface is uniformly emitting.

Figure 5. Left: angles relevant to the path of the emitting spot considered here. The observer’s inclination i is measured from the stellar spin axis Ω. The path on the stellar surface along which the emitting spot moves is described in a coordinate system anchored in the star. The center of the circular path considered here is at colatitude θp and azimuth φp ≡ 0 in this coordinate system. The angular radius of the path is ηs . The azimuth ξs of the spot around the center of the path is measured from its lowest latitude. Right: phase and time residuals of the peak, first harmonic (fundamental), and second harmonic (first overtone) of the observed waveform as functions of the azimuth ξs of a circular spot with a radius of 25◦ moving along a circular path like that shown in the left-hand panel. The phase and time residuals are defined in Section 2.3. In this example, the emission from the spot is assumed to be isotropic, θp = 12◦, ηs = 10◦, i = 45◦, the star has a mass of 1.4M and a radius of 5M , and the star’s spin rate is 400 Hz measured at infinity.

Figure 3. Contour plots of the total fractional rms amplitude produced by isotropic emission from a single stable spot (left-hand panel) and from two stable antipodal spots (right-hand panel), as a function of the stellar radius and the inclination of the primary spot relative to the spin axis, for a 1.4M star with a radius of 5M spinning at 400 Hz observed at an inclination of 45◦. The heavier lines highlight the contours where the fractional flux modulation is 2%. This figure demonstrates that compactness alone cannot explain the low-amplitude oscillations of the AMXPs.

Full text

The Astrophysical Journal, 706:417–435, 2009 November 20 doi:10.1088/0004-637X/706/1/417
C

2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
A MODEL FOR THE WAVEFORM BEHAVIOR OF ACCRETING MILLISECOND X-RAY PULSARS: NEARLY
ALIGNED MAGNETIC FIELDS AND MOVING EMISSION REGIONS
Frederick K. Lamb
1,2
, Stratos Boutloukos
1,3
, Sandor Van Wassenhove
1
, Robert T. Chamberlain
1
,KaHoLo
1
,
Alexander Clare
1
, Wenfei Yu
1
, and M. Coleman Miller
3
1
Center for Theoretical Astrophysics and Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street,
Urbana, IL 61801-3080, USA; fkl@illinois.edu
2
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801-3074, USA
3
Department of Astronomy and Maryland Astronomy Center for Theory and Computation, University of Maryland, College Park, MD 20742-2421, USA
Received 2008 August 31; accepted 2009 October 5; published 2009 October 30
ABSTRACT
We investigate further a model of the accreting millisecond X-ray pulsars we proposed earlier. In this model, the
X-ray-emitting regions of these pulsars are near their spin axes but move. This is to be expected if the magnetic
poles of these stars are close to their spin axes, so that accreting gas is channeled there. As the accretion rate and the
structure of the inner disk vary, gas is channeled along different field lines to different locations on the stellar sur-
face, causing the X-ray-emitting areas to move. We show that this “nearly aligned moving spot model” can explain
many properties of the accreting millisecond X-ray pulsars, including their generally low oscillation amplitudes
and nearly sinusoidal waveforms; the variability of their pulse amplitudes, shapes, and phases; the correlations in
this variability; and the similarity of the accretion- and nuclear-powered pulse shapes and phases in some. It may
also explain why accretion-powered millisecond pulsars are difficult to detect, why some are intermittent, and why
all detected so far are transients. This model can be tested by comparing with observations the waveform changes
it predicts, including the changes with accretion rate.
Key words: pulsars: general – stars: neutron – stars: rotation – X-rays: bursts – X-rays: stars
1. INTRODUCTION
Highly periodic millisecond X-ray oscillations have been de-
tected with high confidence in 22 accreting neutron stars in
low-mass X-ray binary systems (LMXBs), using the Rossi X-
ray Timing Explorer (RXTE) satellite (see Lamb & Boutloukos
2008). We refer to these stars as accreting millisecond X-ray
pulsars (AMXPs). Accretion-powered millisecond oscillations
have so far been detected in 10 AMXPs. They are always ob-
servable in seven AMXPs, but are only intermittently detected
in three others. Nuclear-powered millisecond oscillations have
been detected with high confidence during thermonuclear X-
ray bursts in 16 AMXPs. Persistent accretion-powered mil-
lisecond oscillations have been detected in two AMXPs that
produce nuclear-powered millisecond oscillations; intermittent
accretion-powered millisecond oscillations have been detected
in two others.
The AMXPs have several important properties:
Low oscillation amplitudes. The fractional amplitudes of
the accretion-powered oscillations of most AMXPs are often
only ∼1%–2%.
4
Persistent accretion-powered oscillations with
amplitudes 1% are often detected with high confidence in IGR
J00291+5934 (Galloway et al. 2005; Patruno 2008) and XTE
J1751−305 (Markwardt et al. 2002; Patruno 2008). Persistent
accretion-powered oscillations with amplitudes as low as 2% are
regularly seen in XTE J1807−294 (Zhang et al. 2006; Chou et al.
2008; Patruno 2008), XTE J0929−314 (Galloway et al. 2002),
and XTE J1814−338 (Chung et al. 2008; Patruno 2008). The
4
We characterize the strengths of oscillations by their rms amplitudes
because the rms amplitude can be defined for any waveform, is usually
relatively stable, and is closely related to the power. We convert reported
semi-amplitudes of purely sinusoidal oscillations or Fourier components to
rms amplitudes by dividing by
√
2.
amplitude of the accretion-powered oscillation seen in SWIFT
1756.9−2508 was ∼6% (Krimm et al. 2007). The intermittent
accretion-powered oscillations detected in SAX J1748.9−2021
(Gavriil et al. 2007; Altamirano et al. 2008; Patruno 2008),
HETE J1900.1−2455 (Galloway et al. 2007), and Aql X-1
(Casella et al. 2008) have amplitudes ∼0.5%–3%.
Nearly sinusoidal waveforms. The waveforms (light curves)
of the accretion-powered oscillations of most AMXPs are
nearly sinusoidal (see Wijnands 2006 and the references in
the preceding paragraph). The amplitude of the first harmonic
(fundamental) component is usually 10 times the amplitude of
the second harmonic (first overtone) component, although the
ratio can be as small as ∼3.5, as is sometimes the case in XTE
J1807−294 (Zhang et al. 2006), or even 1, as is sometimes
the case in SAX J1808.4−3658 (see, e.g., Hartman et al. 2008).
Highly variable oscillation amplitudes. The fractional ampli-
tudes of the accretion-powered oscillations of most AMXPs vary
in time by factors ranging from ∼2to∼10. Observed fractional
amplitudes vary from 0.7% to 1.7% in SAX J1748.9−2021,
from 0.7% to 3.7% in XTE J1751−305, from 3% to 7% in XTE
J0929−314, from 1% to 9% in IGR J00291+5934, from 2% to
11% in XTE J1814−338, from 1% to 14% in XTE J1808−338,
and from 2% to 19% in XTE J1807−294 (see the references
above).
Highly variable pulse phases. The phases of accretion-
powered pulses have been seen to vary rapidly by as much as
∼0.3 cycles in several AMXPs, including SAX J1808.4−3658
(Morgan et al. 2003; Hartman et al. 2008) and XTE J1807−294
(Markwardt 2004). Wild changes in the apparent pulse fre-
quency have been observed with both signs at the same accretion
rate in XTE J1807−
294 (see Markwardt 2004). If interpreted as
caused by changes in the stellar spin rate, these phase variations
would be more than a factor of 10 larger than expected for the
largest accretion torques and smallest inertial moments thought
417

418 LAMB ET AL. Vol. 706
possible for these systems (see Ghosh & Lamb 1979b; Lattimer
& Prakash 2001).
Undetected accretion-powered oscillations. More than 80 ac-
creting neutron stars in LMXBs are known (Chakrabarty 2005;
Liu et al. 2007), but accretion-powered millisecond X-ray oscil-
lations have so far been detected in only 10 of them. Accretion-
powered oscillations have not yet been detected even in 13
AMXPs that produce periodic nuclear-powered millisecond os-
cillations, indicating that they have millisecond spin periods
(Lamb & Boutloukos 2008); eight of these also produce kilo-
hertz quasi-periodic oscillations (QPOs) with frequency sepa-
rations that indicate that they not only have millisecond spin
periods but also have dynamically important magnetic fields
(Boutloukos & Lamb 2008).
Intermittent accretion-powered oscillations. Accretion-
powered millisecond X-ray pulsations have been detected
only occasionally in SAX J1748.9−2021 (Gavriil et al. 2007;
Altamirano et al. 2008; Patruno 2008), HETE J1900.1−2455
(Galloway et al. 2007), and Aql X-1 (Casella et al. 2008).
When oscillations are not detected, the upper limits are typically
0.5%.
Correlated pulse arrival times and amplitudes. The phase
residuals of the accretion-powered pulses of several AMXPs
appear to be anti-correlated with their fractional amplitudes, at
least over some of the amplitude ranges observed. AMXPs that
show this type of behavior include XTE J1807−294 and XTE
J1814−338 (Patruno 2008).
Similar accretion- and nuclear-powered pulses. The shapes
and phases of the nuclear-powered X-ray pulses of the AMXPs
SAX J1808.4−3658 (Chakrabarty et al. 2003) and XTE
J1814−338 (Strohmayer et al. 2003) are very similar to the
shapes and phases of their accretion-powered X-ray pulses.
Concentration in transient systems of AMXPs with accretion-
powered oscillations. The AMXPs in which accretion-powered
oscillations have been detected tend to be found in binary
systems that have outbursts lasting about a month (but see
Galloway et al. 2008) separated by quiescent intervals lasting
years (Chakrabarty 2005; Riggio et al. 2008). The accretion
rates of these neutron stars are very low.
In this paper, we investigate further the “nearly aligned
moving spot” model of AMXP X-ray emission that we proposed
previously (Lamb et al. 2006, 2007, 2008). This model has three
main features:
1. The strongest poles of the magnetic fields of neutron
stars with millisecond spin periods are located near—and
sometimes very near—the stellar spin axis. This behavior is
expected for several magnetic field evolution mechanisms.
2. The star’s magnetic field channels accreting gas close
to its spin axis, creating X-ray-emitting areas there and
depositing nuclear fuel there.
5
3. The X-ray-emitting areas on the stellar surface move, as
changes in the accretion rate and the structure of the inner
disk cause accreting gas to be channeled along different
field lines to different locations on the stellar surface. (The
magnetic field of the neutron star is fixed in the stellar crust
on the timescales relevant to the phenomena considered
here.)
5
Muno et al. (2002) considered a single bright spot near the spin axis as well
as a uniformly bright hemisphere and antipodal spots near the spin equator as
possible reasons for the nearly sinusoidal waveforms of some X-ray burst
oscillations, but did not consider accretion-powered oscillations or other
consequences of emission from near the spin axis.
These features provide the basic ingredients needed to under-
stand the AMXP properties discussed above. This is the subject
of the sections that follow. As a guide to these sections, we
summarize our results here.
1. Emission from near the spin axis naturally produces weak
modulation, regardless of the viewing direction. The reason
is that uniform emission from a spot centered on the spin
axis is axisymmetric about the spin axis and therefore
produces no modulation. Emission from a spot close to
the spin axis has only a small asymmetry and therefore
produces only weak modulation.
2. Emission from near the spin axis also naturally produces a
nearly sinusoidal waveform, because the asymmetry of the
emission is weak and broad.
3. If the emitting area is close to the spin axis, even a small
movement in latitude can change the oscillation amplitude
by a substantial factor.
4. If the emitting area is close to the spin axis, movement in
the longitudinal direction by a small distance can change
the phase of the oscillation by a large amount.
Changes in the latitude and longitude of the emitting area
are expected on timescales at least as short as the ∼0.1 ms
dynamical time at the stellar surface and as long as the
∼10 day timescale of the variations observed in the mass
accretion rate.
5. If the emitting area is very close to the spin axis and
remains there, the oscillation amplitude may be so low
that it is undetectable. The effects of rapid changes in
the position of the emitting area—possibly in combination
with other effects, such as reduction of the modulation
fraction by scattering in circumstellar gas—may also play
a role in reducing the detectability of accretion-powered
oscillations in neutron stars with millisecond spin periods.
These effects may explain the fact that accretion-powered
X-ray oscillations have not yet been detected in many
accreting neutron stars that are thought to have millisecond
spin periods and dynamically important magnetic fields.
6. If the emitting area is very close to the spin axis, a small
change in the accretion flow can suddenly channel gas
farther from the spin axis, causing the emitting area to
move away from the axis. This can make a previously un-
detectable oscillation become detectable. Temporary mo-
tion of the emitting area away from the spin axis may
explain the intermittent accretion-powered oscillations of
some AMXPs (Lamb et al. 2009).
7. If the pulse amplitude and phase variations observed in
AMXPs are caused by motion of the emitting area, they
should be correlated. In particular, the pulse phase should
be much more scattered when the pulse amplitude is very
low. The reason is that changes in the longitudinal position
of the emitting area by a given distance produce much larger
phase changes when the emitting area is very close to the
spin axis, which is also when the oscillation amplitude is
very low.
The observational consequences discussed so far depend
only on features (2) and (3) of the model, i.e., that the
accretion-powered X-ray emission of AMXPs comes from
areas near their spin axes and that these areas move
significantly on timescales of hours to days.
8. The picture of AMXP X-ray emission outlined here sug-
gests that the shapes and phases of the nuclear- and
accretion-powered pulses are similar to one another in some
AMXPs because the nuclear- and accretion-powered X-ray

No. 1, 2009 MODEL FOR WAVEFORM BEHAVIOR OF AMXPs 419
emission comes from approximately the same area on the
stellar surface. The reason for this is that in some cases,
the mechanism that concentrates the magnetic flux of the
accreting neutron star near its spin axis, as it is spun up, will
naturally produce magnetic fields strong enough to confine
accreting nuclear fuel near the magnetic poles at least par-
tially, even though the dipole component of the magnetic
field is weak.
9. The picture of neutron star magnetic field evolution and
AMXP X-ray emission outlined here also suggests a pos-
sible explanation for why the AMXPs in which accretion-
powered oscillations have been detected are in transient
systems. If most neutron stars in LMXBs were spun up by
accretion from a low spin rate to a high spin rate, their mag-
netic poles were forced very close to their spin axes, making
accretion-powered oscillations difficult or impossible to de-
tect. However, those stars that are now in compact transient
systems now experience infrequent episodes of mass accre-
tion, and the accretion rate is very low. By now they have
been spun down from their maximum spin rates, a pro-
cess that could force their magnetic poles away from their
spin axes enough to produce detectable accretion-powered
oscillations.
These last two observational consequences depend on feature
(1) of the model, i.e., on how the magnetic fields of neutron
stars evolve as they are spun up and down by accretion and
electromagnetic torques.
In the remainder of this paper, we discuss in detail the features
of the model and its observational implications. In Section 2,we
outline our approach, discussing our modeling of X-ray emis-
sion from the stellar surface, our computational and the code
verification methods, and the pulse profile representation we use.
We present our results in Sections 3 and 4. These results
are based on our computations of several hundred million
waveforms for different emitting regions, beaming patterns,
stellar models, and viewing directions. In Section 3, we consider
the shape and amplitude of X-ray pulses as a function of the size
and inclination of the emitting areas, the compactness of the star,
and the stellar spin rate. In Section 4, we consider the changes
in the pulse amplitude and phase produced by various motions
of the emitting regions on the stellar surface and explore the
origins of correlated changes in the pulse amplitude and phase
and the effects of rapid movement of the emitting areas. We also
discuss why oscillations have not yet been detected in many
accreting neutron stars in LMXBs and why accretion-powered
oscillations are detected only intermittently in some AMXPs.
In Section 5, we summarize the results of our model calcula-
tions. We also discuss how the magnetic poles of most AMXPs
can be forced close to their spin axes, how such mechanisms
may explain why the AMXPs that produce accretion-powered
millisecond oscillations are transient pulsars, the consistency of
the model with the observed properties of rotation-powered mil-
lisecond pulsars, and possible observational tests of the model
discussed here.
Further results of our investigation of the present model
will be presented elsewhere (S. Boutloukos et al. 2009, in
preparation).
2. X-RAY WAVEFORM MODELING
2.1. Modeling the X-ray Emission
In the radiating spot model of AMXP X-ray emission, the
waveform seen by a distant observer depends on the sizes,
shapes, and positions of the emitting regions on the stellar
surface; the beaming pattern of the radiation; the compactness,
radius, and spin rate of the star; and the direction from which the
star is observed. The properties of the X-ray-emitting regions are
determined by the strength and geometry of the star’s magnetic
field, the locations where plasma from the accretion disk enters
the magnetosphere, the extent to which the accreting plasma
becomes threaded and channeled by the stellar magnetic field,
and the resulting plasma flow pattern onto the stellar surface.
In principle, accreting plasma can reach the stellar surface
in two basic ways: (1) by becoming threaded by the stellar
magnetic field and then guided along stellar field lines to the
vicinity of a stellar magnetic pole (Lamb et al. 1973;Basko
& Sunyaev 1975; Elsner & Lamb 1976; Ghosh et al. 1977;
Ghosh & Lamb 1979a, 1979b) or (2) by penetrating between
lines of the stellar magnetic field via the magnetic version of
the Rayleigh–Taylor instability (Lamb 1975a, 1975b; Elsner &
Lamb 1976, 1977; Arons & Lea 1976
;Lamb1977) and then
spiraling inward to the stellar surface.
If a centered dipole component is the strongest component
of the star’s magnetic field, plasma in the accretion disk that
becomes threaded and then channeled to the vicinity of a
magnetic pole is expected to impact the star in a partial or
complete annulus around the pole, producing a crescent- or
ring-shaped emitting area near the pole. If the axis of the dipole
field is significantly tilted relative to the spin axis and the spin
axis is aligned with the axis of the accretion disk, a crescent-
shaped emitting region is expected (see, e.g., Basko & Sunyaev
1975, 1976; Ghosh et al. 1977; Daumerie et al. 1996; Miller
1996; Miller et al. 1998; Romanova et al. 2003). If instead the
dipole axis is very close to the spin axis, as in the model of
AMXP X-ray emission proposed here, the emitting region may
completely encircle the spin axis (see, e.g., Ghosh & Lamb
1979a, 1979b; Romanova et al. 2003).
The north and south magnetic poles of some AMXPs may
be very close to the same spin pole, producing a very off-
center dipole moment orthogonal to the spin axis (see Chen &
Ruderman 1993; Chen et al. 1998; and Section 5.1). If so, ac-
creting matter will be channeled close to the spin axis, but may
be channeled preferentially toward one magnetic pole, produc-
ing an emitting region with approximately onefold symmetry
about the spin axis, or about equally toward both poles, produc-
ing an emitting region with approximately twofold symmetry
about the spin axis. In the first case, the first harmonic of the
spin frequency is likely be the dominant harmonic in the X-ray
waveform, whereas in the second case, the second harmonic
is likely to dominate. Which case occurs will depend on the
accretion flow through the inner disk. In either case, the X-ray
emission will come from close to the spin axis.
The neutron stars that are AMXPs may well have even more
complicated magnetic fields, with significant quadrupole and
octopole components. Higher multipole components are likely
to play a more important role in the AMXPs than in the classic
strong-field accretion-powered pulsars, because the magnetic
fields of the AMXPs are much weaker. As a result, accreting
plasma can penetrate closer to the stellar surface, where the
higher multipole moments of the star’s magnetic field have a
greater influence on the channeling of accreting plasma (Elsner
&Lamb1976). In this case, plasma will still tend to be channeled
toward regions on the surface where the magnetic field is
strongest and will tend to impact the surface in rings or annuli,
but the emission pattern may be spatially complex and vary
rapidly in time (Long et al. 2008).

420 LAMB ET AL. Vol. 706
Disk plasma that penetrates between lines of the stellar
magnetic field will continue to drift inward as it loses its angular
momentum, probably predominantly via its interaction with the
star’s magnetic field (Lamb & Miller 2001). Cold plasma will
remain in the disk plane and impact the star in an annulus where
the disk plane intersects the stellar surface (Miller et al. 1998;
Lamb & Miller 2001). If some of the accreting plasma were
to become hot, the forces exerted on it by the stellar magnetic
field would tend to drive it toward the star’s magnetic equator
(Michel 1977), causing it to impact the stellar surface in an
annulus around the star’s magnetic equator. However, emission
and inverse Compton scattering of radiation is likely to keep the
accreting plasma cold (Elsner & Lamb 1984), so that it remains
in the disk plane as it drifts inward. Plasma that penetrates to
the stellar surface via the magnetic Rayleigh–Taylor instability
is likely to impact the stellar surface in rapidly fluctuating,
irregular patterns (see Romanova et al. 2006, 2008).
Whether accreting plasma reaches the stellar surface pre-
dominantly via channeled flow along field lines or via unstable
flow between field lines depends on the accretion rate and the
spin frequency of the star (see Lamb 1989; Romanova et al.
2008; Kulkarni et al. 2008). Under some conditions, plasma
may accrete in both ways simultaneously (see Miller et al. 1998;
Romanova et al. 2008; Kulkarni et al. 2008).
The sizes, shapes, and locations of the emitting areas on the
surface of an accreting magnetic neutron star and the properties
of the emission from these areas are expected to change on
timescales at least as short as the ∼1 ms dynamical timescale
near the star. This expectation is supported by recent simulations
of accretion onto weakly magnetic neutron stars (see Romanova
et al. 2003, 2004, 2006; Long et al. 2008; Romanova et al. 2008;
Kulkarni et al. 2008). However, changes in AMXP X-ray fluxes
can be measured accurately using current instruments only by
combining ∼100–1000 s of data and hence only variations in
waveforms on timescales longer than this can be measured
directly. Consequently, the emitting areas and beaming patterns
that are relevant for comparisons with current observations of
waveforms are the averages of the actual areas and beaming
patterns over these relatively long times. The emitting areas
and beaming patterns that we use in our computations should
therefore be interpreted as averages of the actual areas and
beaming patterns over these times.
We have computed the X-ray waveforms produced by emit-
ting regions with various sizes, shapes, and positions, for several
different X-ray-beaming patterns and a range of stellar masses,
compactnesses, and spin rates. We find that in many cases these
waveforms can be approximated by the waveforms generated
by a circular, uniformly emitting spot located at the centroid of
the emitting region. The main reason for this is that an observer
sees half the star’s surface at a time (or more, when gravitational
light deflection is included), which diminishes the influence of
the size and detailed shape of the emitting region on the wave-
form. Consequently, we focus here on the waveforms produced
by uniformly emitting circular spots. We will discuss the wave-
forms produced by emitting areas with other shapes, such as
rings or crescents, in a subsequent paper (S. Boutloukos et al.
2009, in preparation).
In addition to studying the X-ray waveforms produced by
emitting areas with fixed sizes, shapes, positions, and radiation-
beaming patterns, we are also interested in the changes in
waveforms produced by changes in the these properties of the
emitting areas. The changes we investigate should be understood
as occurring on the timescales 100 s that can be studied using
current instruments. It is not yet possible to compute from first
principles the accretion flows and X-ray emission of AMXPs
on these timescales, so simplified models must be used. (The
simulations referred to earlier follow the accretion flow for a
few dozen spin periods or dynamical times, intervals that are
orders of magnitude shorter than the intervals that are relevant).
In the following sections, we consider radiation from a single
spot, from two antipodal spots, and from two spots in the same
rotational hemisphere, near the star’s spin axis. Although many
uncertainties remain, recent magnetohydrodynamic simulations
of accretion onto weakly magnetic neutron stars have found that
gas impacts 1%–20% of the stellar surface (Romanova et al.
2004), equivalent to the areas of circular spots with angular
radii of 10
◦
–53
◦
. These radii are consistent with analytical
estimates of the sizes of the emission regions of accreting
neutron stars with weak magnetic fields (Miller et al. 1998;
Psaltis & Chakrabarty 1999). Consequently, we focus on spot
sizes in this range.
An observer may see radiation from a single spot either
because the accretion flow pattern strongly favors one pole of
a dipolar stellar magnetic field over the other, or because the
observer’s view of one pole is blocked by the inner disk or
by accreting plasma in the star’s magnetosphere (see McCray
&Lamb1976; Basko & Sunyaev 1976). An observer may
see radiation from two antipodal spots if emission from both
magnetic poles is visible. Finally, an observer may see radiation
from two spots near the same rotation pole if neutron vortex
motion drives both of the star’s dipolar magnetic poles toward
the same rotation pole (see Chen et al. 1998).
To make it easier for the reader to compare cases, we usually
report results for our “reference” star, which is a 1.4 M

star
with a radius of 5 M in units where G = c = 1 (10.3 km for
M = 1.4 M

), spinning at 400 Hz as measured at infinity, but
we also discuss other stellar models. For the same reason, we
usually consider spots with angular radii of 25
◦
. This is not an
important limitation, because the observed waveform depends
only weakly on the size of the emitting spots, as discussed in
Section 3.2. We describe how the results change if the spot is
larger or smaller.
2.2. Computing X-ray Waveforms
The X-ray waveforms calculated here assume that radiation
propagating from emitting areas on the stellar surface reaches
the observer without interacting with any intervening matter.
The bolometric X-ray waveforms that would be seen by a dis-
tant observer were calculated using the Schwarzschild plus
Doppler (S+D) approximation introduced by Miller & Lamb
(1998). The S+D approximation treats exactly the special rela-
tivistic Doppler effects (such as aberrations and energy shifts)
associated with the rotational motion of the stellar surface,
but treats the star as spherical and uses the Schwarzschild
spacetime to compute the general relativistic redshift, trace the
propagation of light from the stellar surface to the observer,
and calculate light travel-time effects. For the stars considered
here, and indeed for any stars that do not both rotate rapidly
and have very low compactness, the effects of stellar oblate-
ness and frame dragging are minimal and are negligible com-
pared to uncertainties in the X-ray emission (see Cadeau et al.
2007).
We describe the emission from the stellar surface using
coordinates centered on the star. When considering emission
from a single spot, we denote the angle between its centroid

No. 1, 2009 MODEL FOR WAVEFORM BEHAVIOR OF AMXPs 421
and the star’s spin axis by i
s
and its azimuth in the stellar
coordinate system by φ
s
. When considering emission from two
spots, we somewhat arbitrarily identify one as the primary spot
and the other as the secondary spot. We denote the inclination
and azimuth of the centroid of the primary spot by i
s1
and φ
s1
and the inclination and azimuth of the centroid of the secondary
spot by i
s2
and φ
s2
. We denote the inclination of the observer
relative to the stellar spin axis by i.
In computing the waveforms seen by distant observers, we
use as our global coordinate system Schwarzschild coordinates
(r, θ , ϕ, t ) centered on the star with θ = 0 aligned with the
star’s spin axis and ϕ = 0 in the plane containing the spin axis
and the observer. We choose the zero of the Schwarzschild time
coordinate t so that a light pulse that propagates radially from
a point on the stellar surface immediately below the observer
(i.e., at θ = i and ϕ = 0) arrives at the observer at t = 0.
We carried out many calculations to test and verify the
computer code used to obtain the results we report here. We
determined that the code was giving sufficiently accurate results
by varying the spatial and angular resolutions used. For most
of the cases considered in this paper, the emitting spots were
sampled by a grid of 250 points in latitude and 250 points in
longitude, the radiation-beaming pattern was specified at 10
4
angles, and the flux seen by a distant observer was computed at
10
4
equally spaced values of the star’s rotational phase. In some
cases, finer grids were used.
We verified the code used here by comparing its results with
analytical and numerical results for several test cases:
1. We tested our code’s representation of emitting areas and
ray tracing in flat space by comparing the results given by
our code with exact analytical results for the absolute flux
seen by an observer directly above uniform, isotropically
emitting circular spots of various sizes. The numerical
results agreed with the analytical results.
2. We tested our code’s computation of special relativistic
Doppler boosts, aberrations, and propagation-time effects
in several ways. We compared the results given by our code
with exact analytical results for the waveforms produced
by emission in (a) a pencil beam normal to the surface
and (b) a thin fan beam tangent to the surface of a rapidly
rotating star. We also compared the results given by our
code with analytical results for the waveforms produced
by a small spot on the surface of a slowly rotating star in
flat space emitting (a) isotropically and (b) in a beaming
pattern representing Comptonized emission (see Poutanen
&Gierli
´
nski 2003). The numerical results agreed with the
analytical results.
3. We tested our code’s computation of the general relativistic
redshift and light deflection for nonrotating stars by (a)
comparing the deflection of a fan beam tangent to the
stellar surface given by our code for a variety of stellar
compactnesses with the analytical expressions for the light
deflection given by Pechenick et al. (1983) and Page
(1995); (b) comparing the absolute flux given by our
code for an observer directly above isotropically emitting
uniform circular spots of various sizes with independent
semi-analytical results for these cases; (c) comparing the
symmetries of the waveform and the phase of the waveform
maximum given by our code with exact analytical results
for these quantities; and (d) comparing the shape of the
waveforms given by our code with the shapes reported
by Pechenick et al. (1983) and Strohmayer (1992). Our
numerical results agreed satisfactorily with the comparison
results in all cases.
6
4. We tested our code’s computation of the waveforms pro-
duced by emission from slowly rotating stars in general
relativity by comparing the rms oscillation amplitudes it
gives with the amplitudes given by the approximate analyt-
ical formulae of Viironen & Poutanen (2004). Where the
results of Viironen & Poutanen are expected to be accurate,
the two sets of rms amplitudes agreed to better than 1%; in
many cases the agreement was much better. We also com-
pared the waveform given by our code for an isotropically
emitting spot inclined 45
◦
from the spin axis of a 1.4 M

star with a Schwarzschild coordinate radius of 5M spinning
at 600 Hz seen by an observer at an inclination of 45
◦
with
the waveform reported by Poutanen & Beloborodov (2006)
for this case; the two waveforms agreed to better than 1%.
Further details of these tests and comparisons will be given in
a subsequent paper (S. Boutloukos et al. 2009, in preparation).
2.3. Constructing Pulse Profiles
The X-ray flux seen by a given observer will evolve continu-
ously in time as the star rotates and the emission from the stellar
surface changes, generating the observed waveform W (t). As
noted in Section 2.1, the accretion flow from the inner disk to
the stellar surface is expected to vary on timescales at least as
short as the ∼1 ms dynamical timescale near the neutron star,
which will cause the sizes, shapes, and positions of the emitting
regions, and therefore the observed waveform, to vary on these
timescales.
The sensitivity of current instruments is too low to measure
the waveform of an AMXP on timescales as short as 1 ms.
However, nearly periodic waveforms with periods this short
can be partially characterized by folding segments of flux data
centered at a sequence of clock times t
i
(see, e.g., Hartman et al.
2008; Patruno 2008). If the data are folded with a period P
f
that is chosen to agree as closely as possible with the local,
approximate repetition period P (t
i
) of the waveform, one can
construct a time sequence of pulse profiles W
P
(φ,t
i
); here φ is
the pulse phase over one cycle.
The pulse profiles W
P
(φ,t
i
) constructed by folding flux data
are averages of the actual pulse profiles over the time interval
required to construct a stable profile, which can be hundreds
or even thousands of seconds, 10
5
–10
6
times longer than the
∼1 ms dynamical timescale near the neutron star. The folded
pulse profiles are therefore likely to vary more slowly and have
less detail than the X-ray waveform, a point to which we will
return in Section 4.
The waveforms of AMXPs can be modeled even on the
dynamical timescale near the stellar surface, but such waveforms
would contain much more information than can be studied using
current observations. Consequently, we focus here on modeling
folded pulse profiles. We define a computed pulse profile as the
waveform seen by a given observer when a star with a constant
6
The waveforms reported by Pechenick et al. (1983) for two antipodal spots
are slightly inaccurate, as shown by the following two tests. The waveform
seen by an observer in the star’s rotation equator viewing two identical
antipodal spots in the rotation equator should be the same at 180
◦
as at 0
◦
,but
this is not the case for their waveform for this geometry (see their Figure 7).
More generally, the flux from a uniform, isotropically emitting, circular spot
on a spherical, nonrotating star should depend only on the angle between the
radius through the center of the spot and the radius to the observer. This is not
quite true for the waveforms reported by Pechenick et al. (1983). The
waveforms given by our code pass these tests (for details, see S. Boutloukos
et al. 2009, in preparation).

Frequently asked questions

Q1. What are the contributions in "C: " ?

The authors investigate further a model of the accreting millisecond X-ray pulsars they proposed earlier. The authors show that this “ nearly aligned moving spot model ” can explain many properties of the accreting millisecond X-ray pulsars, including their generally low oscillation amplitudes and nearly sinusoidal waveforms ; the variability of their pulse amplitudes, shapes, and phases ; the correlations in this variability ; and the similarity of the accretionand nuclear-powered pulse shapes and phases in some. 

Q2. What are the future works in "C: " ?

After the Workshop, Watts et al. ( 2008 ) investigated this possibility and found just such a correlation in the RXTE data on XTE J1814−338. This indicates that, as the authors had suggested, the accretion- and nuclear-powered emitting regions in this pulsar very nearly coincide, and that the simultaneous wandering of the arrival times of both oscillations by ∼1 ms ( ∼0. 3 in phase ) during the outburst is due to wandering of the matter ( and hence the fuel ) deposition pattern on the stellar surface.