A model for the waveform behavior of accreting millisecond pulsars: Nearly aligned magnetic fields and moving emission regions
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 (8 min read)
- 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.
4.4. 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.
- 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.
- 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.
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Frequently Asked Questions (2)
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.