A MODEL FOR THE WAVEFORM BEHAVIOR OF ACCRETING MILLISECOND X-RAY PULSARS: NEARLY ALIGNED MAGNETIC FIELDS AND MOVING EMISSION REGIONS

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; Lamb 1977) 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, accreting matter will be channeled close to the spin axis, but may be channeled preferentially toward one magnetic pole, producing an emitting region with approximately onefold symmetry about the spin axis, or about equally toward both poles, producing 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 & Lamb 1976). 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).

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 predominantly 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 emitting 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 waveform. Consequently, we focus here on the waveforms produced by uniformly emitting circular spots. We will discuss the waveforms 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 & Lamb 1976; 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.

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 distant observer were calculated using the Schwarzschild plus Doppler (S+D) approximation introduced by Miller & Lamb (1998). The S+D approximation treats exactly the special relativistic 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 oblateness and frame dragging are minimal and are negligible compared 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 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 10⁴ angles, and the flux seen by a distant observer was computed at 10⁴ 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 & Gierliński 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 produced by emission from slowly rotating stars in general relativity by comparing the rms oscillation amplitudes it gives with the amplitudes given by the approximate analytical 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 compared 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).

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.

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, 10⁵–10⁶ 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 emission pattern makes one rotation. It is customary and useful to describe pulse profiles by the amplitudes and phases of their Fourier components. We describe our computed profiles using the representation

$$\begin{equation} W_P(t) = A_{0} + \sum _{k=1}^{\infty } A_k \cos [2\pi k (\nu _s t - \phi _k)] \;. \end{equation} \tag{ 1 }$$

Here t is the time measured at infinity, A₀ is the average of the flux over one rotation period, A_k and ϕ_k are the amplitude and phase of the kth harmonic component of the pulse, and ν_s is the stellar rotation frequency. For harmonic k, the range of unique phases is 0 to 1/k. We sometimes also refer to the phase ϕ_p of the peak of the pulse profile, defined in a similar way. With these definitions, shifting the arrival time of a given pulse by an amount δt shifts the phases of its Fourier components and the phase of the pulse peak by ν_s δt cycles.

We define the arrival time t_k of the kth harmonic component of the pulse profile to be zero if the first maximum of the sinusoid associated with this component reaches the observer at the same time as would a light pulse coming from a point on the stellar surface immediately below the observer (see Section 2.2). The arrival time t_k is affected by the sizes, shapes, and locations of the emitting areas, the beaming pattern of the emitted radiation, and the aberration and Doppler shifts produced by the spin of the star and usually is not zero. We define the arrival phase of the kth harmonic component by ϕ_k = ν_s t_k. The arrival time t_p and arrival phase ϕ_p of the peak of the pulse profile are defined similarly. We refer to t_k and t_p as time residuals and to ϕ_k and ϕ_p as phase residuals. An increase in a time or phase residual indicates that the harmonic component or the peak is arriving later.

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 angular pattern of the radiation flux produced by Thomson scattering is strongly peaked in the direction normal to the stellar surface, whereas the flux patterns produced by fan-shaped radiation beams like those proposed by Poutanen & Gierliński (2003) to describe Comptonized emission from near the stellar surface are peaked much less (if at all) in the direction normal to the stellar surface. The angular pattern of the radiation flux produced by isotropic emission is intermediate between these two cases, being peaked normal to the surface less than the pattern produced by Thomson scattering but more than the pattern produced by fan-beam emission. We first discuss the waveforms produced by isotropic emission and then the waveforms produced by other beaming patterns.

Figure 1 shows the fractional rms amplitudes of the first harmonic (fundamental) and second harmonic (first overtone) components of the pulse profile and the total fractional rms amplitude produced by isotropic emission from a single stable spot and from two stable antipodal spots, as a function of the observer's inclination and the inclination of the primary spot relative to the spin axis. The curves are for spots with angular radii of 25° on our reference star (a 1.4 M_☉ star with a radius of 5 M spinning at 400 Hz). As expected, all amplitudes are zero when the star is viewed along its spin axis (i = 0°) or the emitting spots are centered on the rotation pole (i_s = 0° or i_s1 = 0° and i_s2 = 180°). In addition, the amplitude of the first harmonic variation produced by two stable antipodal spots vanishes if either the observer or the spots are in the rotation equator (i_s = 90° or i_s1 = i_s2 = 90°), whereas the amplitude of the second harmonic is highest for these geometries.

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The top panels in Figure 1 show that if the observer sees radiation only from a single spot, the total fractional amplitude can be as low as ∼1%–2% for a range of viewing angles only if the spot is within a few degrees of the spin axis. These panels also show that for radiation from a single spot, the second harmonic is ∼5–10 times smaller that the first harmonic for most observer and spot inclinations.

The bottom panels in Figure 1 show that if the observer sees radiation from two antipodal spots, the total fractional amplitude can be as low as ∼2% for a substantial range of viewing angles only if the spots are within about 5° of the spin axis. These panels also show that for radiation from two antipodal spots, the second harmonic is smaller than the first harmonic for observer or spot inclinations less than about 60°. The amplitude of the second harmonic is ≲2% for all observers only if the spots are within 15° of the spin axis.

Two emitting spots centered near the same rotation pole are expected if the inward motion of neutron vortices in the stellar core drives both of the star's magnetic poles toward the same rotation pole (see Section 5.1). The fractional modulation produced by two spots with this geometry is typically somewhat larger than the modulation produced by two antipodal spots if the observer and spot inclinations are both large, but is typically smaller than the modulation produced by two antipodal spots if the observer inclination or the spot inclination is small. Even so, the total fractional amplitude can be as low as ∼1%–2% for a range of viewing angles only if the spots are within a few degrees of the stellar spin axis. For example, for our reference star the fractional modulation produced by two spots at the same inclination but separated by 160° in longitude is larger than the modulation produced by two antipodal spots if i ≳ 70° and i_s1 = i_s2 ≳ 70° but is smaller if i ≲ 60° and i_s1 = i_s2 ≲ 60°. The total fractional amplitude can appear as low as ∼1%–2% over half the sky (i ⩾ 60°) only if the two spots are within ∼10° of the spin axis.

The intensity distribution produced by a Thomson scattering atmosphere is described by a linear combination of Hopf functions (see Chadrasekhar 1960, Section 68) and peaks in the direction normal to the stellar surface. The radiation flux from such an atmosphere is given by the product of this intensity distribution and the projected area of the surface, which is proportional to the cosine of the angle to the normal. Consequently, the radiation flux from a Thomson scattering surface is strongly peaked in the direction normal to the surface. For this reason, emission from a Thomson scattering region typically produces a higher modulation fraction than would isotropic emission from the same region. Hence if Thomson scattering is important, the emitting regions must be closer to the star's spin axis than if the emission is isotropic.

Fan-shaped intensity distributions like those proposed by Poutanen & Gierliński (2003) to describe Comptonized emission from near the stellar surface can partially compensate for the decrease of the projected area as the angle from the normal to the surface increases, producing a radiation flux distribution that is more isotropic than that produced by an isotropic intensity distribution. For this reason, fan-shaped emission from a single spot can produce modulation at the first harmonic of the spin frequency that is smaller than would be produced by isotropic emission from the same spot. A spot producing this emission pattern must still be close to the spin axis in order for the amplitude of the first harmonic to be ≲2%. Fan-shaped emission from a single spot tends to increase the amplitude of the second harmonic relative to the first. Fan-like emission from two antipodal spots produces waveforms similar to those produced by isotropic emission, but the modulation fraction can be up to a factor of 2 larger.

In SAX J1808.4−3658, the second harmonic is usually weak but is occasionally as strong as or even stronger than the first harmonic (see Hartman et al. 2008). The strengthening of the second harmonic could be caused by development of a twofold asymmetry of one spot, by more fan-beam emission from a single spot, or by increased visibility a second, antipodal spot.

Our computations show that modulation fractions are below typical current detection limits if the emitting regions are within 1° of the spin axis and remain there. As discussed in Section 4, rapid fluctuations in the amplitude and phase of the oscillations would not be directly detectable in time-averaged observations, but would raise the background noise level and further reduce the detectability of the oscillations. These effects, possibly in combination with others, such as reduction of the modulation fraction by scattering in circumstellar gas (Lamb et al. 1985; Miller 2000), may explain the nondetection of accretion-powered oscillations in accreting neutron stars in which nuclear-powered oscillations have been detected.

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. This is illustrated in Figure 2, which shows the total fractional amplitude produced by isotropic emission from a single spot and from two antipodal spots, as a function of the radius of the spots and the inclination of the primary spot relative to the spin axis. All the curves in this figure are for our reference star, viewed at an inclination of 45°.

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The left-hand panel of Figure 2 shows that even a single spot that covers half of the star (a spot with a radius of 90°) can produce a total fractional modulation as small as 2% only if its inclination is ≲5°. The modulation fraction depends only weakly on the size of the spot, especially for small-to-moderate spot inclinations. For example, 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°. Even for a spot inclined 90° from the spin axis, the fractional flux modulation depends only weakly on the size of the spot for spot radii ≲50°, decreasing from 70% to 35% as the spot radius increases from 30° to 90°. The reason for this insensitivity is that the angular width of the stellar radiation pattern produced even by spots with radii ∼50° is governed mostly by the beaming pattern of the radiation from the stellar surface and the magnitude of the gravitational light deflection, not by the radius of the spot. A single spot at an inclination of 70° can produce a fractional modulation ≲2% only if the entire surface of the star is uniformly emitting except for a dark area of radius 20° on the opposite side of the star from the center of the spot (i.e., if the spot radius is ≳160°).

The right-hand panel of Figure 2 shows that two uniformly emitting antipodal spots of a given radius produce a smaller fractional modulation than a single spot of the same radius, because the antipodal spots cover twice as much of the stellar surface. Even so, the antipodal spots must be very large in order to produce fractional modulations as small as those observed. For example, two antipodal spots on our 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°. The fractional modulation for most observing directions doubles to ∼4% if the spot radius decreases from 75° to 65°.

These results show that the low fractional amplitudes often observed in the AMXPs cannot be explained by large emitting regions, unless the emitting regions are uniform and cover almost the entire stellar surface, which is not expected. 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. Hence attributing their small amplitudes at some times to large emitting areas would require their emitting areas to shrink by a large factor at other times, which is not expected.

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). The reason is that gravitational focusing allows the observer to see a larger fraction of the surface of such a star, averaging the contributions of its bright and dark areas and reducing the flux variation seen as the star turns. Strong gravitational focusing can increase the fractional modulation seen by some observers if the star is very compact (R ≲ 3.5 M).

Even if all AMXPs are relatively compact (R ∼ 4 M), which is not expected, they would produce fractional modulations larger than the ∼1%–2% fractional modulations often observed (see Section 1) unless their emitting regions are close to their stellar spin axes. Nor can the failure so far to detect accretion-powered oscillations in some nuclear-powered AMXPs be explained unless the accretion-powered emission comes from areas very close to the stellar spin axis. These points are illustrated by Figure 3, which shows the fractional amplitudes produced by isotropic emission from a single spot (left-hand panel) and from two antipodal spots (right-hand panel) as a function of R/M and the inclination of the primary spot relative to the spin axis, for a 1.4 M_☉ star spinning at 400 Hz observed at an inclination of 45°.

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The left-hand panel of Figure 3 shows that even if all the AMXPs had radii as small as 4 M, their fractional modulations could be as small as ∼2% for emission from a single spot only if the spot is within 3° of the spin axis. If instead the spot is at a moderate-to-high inclination, their fractional modulations would be much larger. For example, if the spot is 45° from the spin axis, the fractional modulation would be ∼25%.

The right-hand panel of Figure 3 shows that although the fractional modulation produced by two antipodal spots is generally smaller than the modulation produced by a single spot, antipodal spots would have to be within 10° of the spin axis to produce a fractional modulation as small as ∼2%, even for a star as compact as 4 M. If instead the spots are 45°, from the spin axis, the fractional modulation would be ∼7%, much larger than the lowest modulations observed in most AMXPs.

As another example (not plotted here), consider a relatively massive 2.2 M_☉ star with one or two spots that are not close to the spin axis, observed at an inclination of 45°. If there is a single, isotropically emitting spot of radius of 25° at an inclination of 45°, the fractional modulation would be ∼30% if the star has a radius of 4 M and 22% if the star has a radius of 3.2 M. If instead there are two antipodal, isotropically emitting spots at inclinations of 45° and 135°, the fractional modulation would be ∼11% if the star has a radius of 4 M and 9% if the star has a radius of 3.2 M. These fractional modulations are much larger than the lowest modulations observed in most AMXPs. The fractional modulation is typically even larger for stars that are still more compact, because of the strong gravitational focusing effect discussed above.

These results show that even if the AMXPs are compact neutron stars, their fractional modulation will be much larger than the low fractional modulations they often produce, unless their emitting areas are within a few degrees of the spin axis. Hence the low fractional modulations often observed in the AMXPs cannot be attributed to high stellar compactness.

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. The compactness of the neutron star cannot change significantly on such short timescales, and hence some mechanism other than high compactness must be responsible for the low fractional modulations seen at many times in most AMXPs.

Our results also show that high stellar compactness cannot be the main factor responsible for the nondetection of accretion-powered oscillations in many accreting neutron stars in low-mass X-ray binary systems, including many stars in which nuclear-powered oscillations have been detected. The upper limits on the fractional amplitude of any accretion-powered oscillation produced by these stars are ≲0.5% (see Section 1), much less than could be explained by the effects of high stellar compactness.

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. However, the dependence of the fractional amplitude on the stellar spin rate is weak. Consequently, the basic conclusions reached in this section are valid for the full range of AMXP spin rates observed.

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. These curves are for spots with angular radii of 25° on our reference star.

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The results shown in Figure 4 support our conclusion in Section 3 that the oscillation amplitudes seen by most observers can be as small as ∼1%–2% only if the emitting regions are within a few degrees of the spin axis. If instead the emitting regions were ≳45° from the spin axis, most observers would see modulation fractions ≳15%.

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 reason is that the star's radiation pattern becomes steadily more asymmetric about the spin axis as the emitting region moves farther from the spin axis, partly because the emitting region becomes more asymmetric and partly because the surface velocity becomes higher.

If the emitting area is initially close to the spin axis, a modest increase or decrease in its latitude can change the amplitude of the observed flux oscillation by a substantial factor. For emission from a single spot, the amplitude increase is steepest for observers at high inclinations, whereas for emission from two antipodal spots, the amplitude increase is steepest for observers at moderate inclinations.

These effects are illustrated by the left-hand panel of Figure 4. It shows that for a single spot initially at i_s = 10°, an increase of the spot inclination by as little as ∼5° can increase the fractional amplitude of the oscillations seen by an observer at 45° by a factor of 2. For observers at higher inclinations, the oscillation amplitude increases even more.

A single spot at a high inclination produces oscillations with a high fractional amplitude for observers at inclinations ≳60° and the amplitude increase only slightly as the inclination increases further. As an example, for the parameter values assumed in Figure 4, the total fractional amplitude seen by an observer at i = 45° increases steadily with increasing spot inclination, reaching ∼70% when i_s = 90°. In contrast, the fractional amplitude seen by an observer at i ≈ 90° (i.e., close to the spin equator) is ∼85% for i_s = 70° and ∼90% for i_s = 90°.

If instead the observer sees radiation from two antipodal spots, the total fractional amplitude increases with spot inclination for spot inclinations ≲50°, though not as steeply as for a single spot. The fractional amplitude eventually decreases with increasing spot inclination, unless the observer is located in the spin equator. A change in the inclination of two antipodal spots by 10° can change the total fractional amplitude seen by an observer at 45° by a factor of 2.

The right-hand panel of Figure 4 shows that the fractional amplitude of the second harmonic is almost the same for radiation from one spot or from two antipodal spots, independent of the inclinations of the observer and the spot(s). Comparison of the right-hand panel of Figure 4 with the left-hand panel shows that the second harmonic dominates the pulse profile for spot inclinations ≲80°.

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. 2008; Patruno 2008; Zhang et al. 2006; Chou et al. 2008). Figure 4 shows that amplitudes this large are possible for isotropic emission from a single spot even for relatively small spot inclinations, provided the observer's line of sight is not too close to the star's spin axis. For example, amplitudes this large are possible for spot inclinations ∼5°–15° if the observer's inclination is ≳50°. A decrease in the spot inclination to ∼1°–3° would then reduce the fractional amplitude to ∼1%–3%, the lowest values observed in these AMXPs.

If the emitting areas of the AMXPs are typically located at high stellar latitudes, as in the model discussed here, their pulse amplitudes are expected to depend on the accretion rate and structure of the inner disk. These may vary on timescales as short or shorter than the ∼0.1 ms dynamical time at the stellar surface up to timescales as long as the ∼10 day observed variations of the mass accretion rate. Consequently, the harmonic amplitudes of AMXP pulses are expected to vary on these timescales.

This model can be tested on timescales longer than the ∼10²–10³ s integration times required to construct pulse profiles by searching for correlations between the amplitudes of the harmonic components of the pulse profile or the variance of the pulse amplitude and the pulsar's X-ray flux or spectrum. Indeed, the rms fractional amplitudes of the pulses of XTE J1814−338 appear to be positively correlated with its mean X-ray flux (Chung et al. 2008). Methods for testing the model on timescales shorter than the time required to construct a pulse profile are discussed in Section 4.5.

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. Obviously, the phase of the pulse peak is also shifted by this amount. If the emitting area is close to the spin axis, a displacement in the azimuthal direction by even a small distance can produce large phase shifts.

A change in the latitude (inclination) of the emitting area not only alters the amplitudes of the Fourier components of the pulse but also their arrival times (phases). These alterations change the shape and arrival time of the pulse in a complex way. They occur because a change in the latitude of the emitting area changes the velocity of the emitting surface, altering the aberration, Doppler shift, and propagation time of each photon. The resulting changes in the pulse shape and arrival time depend on the beaming pattern of the radiation, the properties of the star, and the inclination of the observer. The phase shifts caused by changes in the latitude of the emitting area are expected to be modest, because the surface velocities of neutron star models with spin frequencies ≲600 Hz are typically a small fraction of the speed of light.

Figure 5 shows how the phases of the peak of the pulse and its first and second harmonic components vary as the emitting spot moves along a circular path on the stellar surface. Motion along this path changes both the latitude and the longitude of the spot. Motion in either direction along all or part of such a path may occur if the main impact area of the accreting matter moves around the magnetic pole as a result of changes in the structure of the inner disk, changes that could be caused by variations of the accretion rate onto the star (see Section 2.1).

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The left-hand panel of Figure 5 introduces the angles used to describe the motion of the spot around the path. The right-hand panel shows the changes in the arrival times and phases of the peak of the pulse and its Fourier components that occur as the spot moves around the path shown in the left-hand panel. In this example, the phases of the peak and the first and second harmonic components vary by ∼0.35 cycles, which is comparable to the phase variations seen in XTE J1807−294 (Markwardt 2004) and SAX J1808.4−3658 (Morgan et al. 2003; Hartman et al. 2008) on timescales of hours to days. If the path of the emitting region approaches or loops the spin axis, even a very small displacement can shift the phases by more than one cycle. The phase shifts shown in Figure 5 are typical for isotropic emission. The phase shifts for Comptonized-emission-beaming patterns like those proposed by Poutanen & Gierliński (2003) are similar, but differ in detail.

The phase and time residuals shown in Figure 5 are not zero when the spot is in the plane defined by the spin axis and the observer (ξ_s = 0) for two reasons. First, when ξ_s = 0 the spot is not directly beneath the observer in this example, so a photon emitted at this moment will reach the observer later than would a photon emitted from the point directly beneath the observer, which is the emission point that defines the zero of the arrival time at the observer (see Section 2.3). Second, the spinning motion of the stellar surface produces aberrations, time delays, and Doppler energy shifts that distort the pulse, causing its harmonic components to reach the observer at times different from their arrival times from a nonrotating star. In this particular example, these effects cause the peak of the pulse and its harmonic components to arrive earlier than they would if the star were not rotating. The phase advance produced by these effects more than offsets the backward phase shift produced by the extra travel time.

As the emitting spot moves around the path shown in Figure 5, its inclination increases from 2° to 22°. This causes the amplitude of the oscillation to increase from ∼1% when the spot is closest to the rotation pole to ∼15% when it is farthest from the pole. This variation is comparable to the amplitude variations observed in XTE J1814−338 and XTE J1807−294 (Chung et al. 2008; Patruno 2008; Zhang et al. 2006; Chou et al. 2008).

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. In general, an increase in the inclination of an emitting spot by 20° can by itself shift the phases of the Fourier components of the pulse by amounts ranging from ∼0.05 to ∼0.20 cycles, depending on the beaming pattern, stellar properties, and viewing direction, and can shift the phases of different harmonic components by different amounts. These effects are included in the phase shifts shown in Figure 5.

These results show that if the emitting areas of the AMXPs are usually near their spin axes, as in the model discussed here, movement of the emitting area by modest distances can cause the arrival times of pulse Fourier components to vary by the large amounts observed, even if the AMXPs spin at a nearly constant rate.

Emitting areas occur where accreting plasma impacts the stellar surface. As discussed in Section 2.1, the position of these areas depends on how accreting plasma enters the magnetosphere and hence on the accretion rate and the structure of the inner disk. Consequently, pulse phase variations produced by changes in the location of the emitting areas should be correlated with changes in the luminosity and spectrum of the pulsar. The properties of the accretion flow are expected to vary on timescales ranging from as short as ∼0.1 ms to as long as ∼10 days, and the arrival times of pulse Fourier components should therefore vary on these timescales also.

This model of pulse phase variations can be tested on timescales longer than the ∼10²–10³ s times required to construct a pulse profile by searching for and studying correlations between the measured phases of the harmonic components of pulses or the variance of these phases and the pulsar's X-ray flux or spectrum. 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. Riggio et al. (2008) have reported a strong anticorrelation between the X-ray flux of XTE J1807−294 and the phase residual of the second harmonic component of its pulse profile. Methods for testing this model of phase variations on timescales shorter than the time required to construct a pulse profile are discussed in Section 4.5.

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.

A strong expectation in this model is that the scatter of the pulse arrival times (i.e., the scatter of the time or phase residuals) should decrease steeply with increasing pulse amplitude. The reason for this is that, as shown previously, a given variation in the azimuthal position of the emitting area produces a much larger phase shift when the emitting area is very close to the spin axis than when it is farther away; in contrast, the oscillation amplitude is much smaller when the emitting area is very close to the spin axis than when it is farther away.

This effect is illustrated in Figure 6, which shows the expected distribution of pulses in the arrival time versus amplitude plane if they are produced by emission from an area close to the star's spin axis that wanders over time. 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. The pulse phase residuals seen in several AMXPs (see, e.g., XTE J1807−294: Patruno 2008; SAX J1748.9-2021: Patruno 2008; and IGR J00291+5934: Patruno 2008) have distributions similar to that shown in Figure 6. These distributions are consistent with the model discussed here if the emitting areas of these AMXPs wander in the azimuthal direction by distances equal to ∼10⁻²–10⁻³ of the stellar circumference.

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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. Such a track will be formed if the emitting areas where the accreting matter impacts the stellar surface tend to move along a favored path in stellar latitude and longitude as the accretion rate and the structure of the inner disk change. This is expected because models of the flow of accreting matter from the inner disk to the stellar surface predict that matter will be guided along particular but different magnetic flux tubes as the accretion rate and the structure of the inner disk vary (see Section 2.1).

Our computations show that motion of the emitting area along a path that changes its latitude but not its longitude can shift the phases of the first and second harmonics by at least 0.15 cycles and by different amounts. Motion of the emitting area along a path that instead changes its longitude but not its latitude can shift the phases of the first and second harmonics and the pulse peak by much larger amounts, if the emitting area is near the spin axis, but in this case the phase shifts will be the same for all Fourier components. Only if the shift in pulse phase produced by the change in the longitude of the emitting area exactly compensates for the shift produced by the change in the latitude of the emitting area will there be no trend in the pulse phase residuals with increasing pulse amplitude.

No track may be discernible in the phase-residual versus pulse-amplitude plane when the pulse amplitude is small because, as has just been discussed, the pulse phase is very sensitive to the position of the emitting area when the area is near the spin axis, which is where it is expected to be when the pulse amplitude is small. If, however, the emitting area moves well away from the spin axis, which in the moving spot model produces a substantial increase in the pulse amplitude, a track may become apparent. Tracks are observed in the data on several AMXPs when the pulse amplitude is ≳3% (Patruno 2008). The phase of the first harmonic component of the pulse is correlated with the amplitude of the pulse in IGR J00291+5934 and XTE J1751−305 but anticorrelated in XTE J1807−294 and XTE J1814−338.

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. However, the path along which the emitting area moves in the moving spot model can be inferred from simultaneous measurements of the phases and amplitudes of pulses.

Similar shifts in the phases of all Fourier components are consistent with movement of an emitting area of roughly fixed size, shape, and radiation-beaming pattern along a path that causes a substantial change in its longitude as its latitude decreases. This behavior is observed in XTE J1814−338 (Papitto et al. 2007; Chung et al. 2008; Patruno 2008). 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. This behavior is observed in XTE J1807−294 (Patruno 2008).

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.

This interpretation can be tested by comparing the observed variations of the amplitudes and phases of the Fourier components of the accretion- and nuclear-powered oscillations with the variations predicted by this model. 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, we 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. The final configuration of the magnetic field produced by this mechanism could have a total strength ∼10¹¹–10¹²G, but a dipole component ∼10⁸ G, consistent with the dipole moments inferred from the X-ray emission and spin evolution of AMXPs and millisecond radio pulsars (Lamb & Boutloukos 2008).

Surface magnetic fields ∼10¹¹–10¹² G are strong enough to laterally confine accreting matter within the surface layers of the neutron star, creating an accumulation point for fuel for thermonuclear bursts and causing thermonuclear burning to be concentrated near the star's magnetic poles (Woosley & Wallace 1982; see also Melatos & Phinney 2001 and references therein). They are also strong enough to produce strong-magnetic-field features in the keV X-ray spectra of AMXPs (see Mészáros 1992). If this picture is correct, such features are more likely to be detected in AMXPs such as SAX J1808.4−3658 and XTE J1814−338 that produce nuclear-powered oscillations that are phase-synchronized (or nearly synchronized) with their accretion-powered oscillations than in other AMXPs.

Our 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. Pulse phase and shape variations are therefore expected on this timescale.

The observable effects of changes in the position of the emitting area on timescales longer than the ∼10²–10³ s integration times required to construct a pulse profile have been discussed in previous sections. Here we 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.

Changes in the position of the emitting area on timescales shorter than the time required to construct a pulse profile will appear as unresolved amplitude and phase noise (see Lamb et al. 1985). This noise will reduce the apparent fractional amplitudes of the oscillations at the spin frequency and its harmonics, making them appear weaker than in the plots presented in this paper, which assume stable emitting spots fixed on the stellar surface. 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 trend will tend to reduce further the apparent pulse amplitude when the emitting area is close to the spin axis, making it appear even smaller relative to the pulse amplitude when the emitting area is further from the spin axis than is shown in the plots presented earlier in this section.

Pulse shape fluctuations caused by rapid changes in the position of the emitting region will appear as background noise in excess of the Poisson counting noise. 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. Detection of more excess noise when the apparent pulse amplitude is smaller would support the moving spot model of pulse phase and amplitude variations.

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). Taken together, these effects may help explain the nondetection of accretion-powered oscillations in some accreting neutron stars in which nuclear-powered oscillations have been detected.

Location of the X-ray-emitting areas close to the spin axis also suggests a natural explanation for the sudden appearance and disappearance of accretion-powered oscillations observed in the "intermittent" AMXPs (see Lamb et al. 2008; Boutloukos et al. 2008). 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. Conversely, a change in the accretion flow that suddenly channels accreting gas closer to the spin axis could make a detectable oscillation undetectable. As noted by Lamb et al. (2008), this idea can be tested by studying short-term variations in the amplitudes and phases of the harmonic components of the pulses.

The neutron vortices in the fluid core of a spinning neutron star are expected to move radially inward if the star is spun up. This inward vortex motion is expected to drag the magnetic flux tubes in the fluid core of the star toward the star's spin axis (Srinivasan et al. 1990; Ruderman 1991).⁷ Some of the relevant physics is not well understood and many details remain to be explored, but this process is expected to squeeze the magnetic flux of accreting pulsars toward their spin axes as they are spun up to short periods (recycled). As a result, the magnetic poles of recycled millisecond pulsars are expected to be very close to their spin axes (Chen & Ruderman 1993; Chen et al. 1998). The magnetic poles of recycled pulsars could also end up close to their spin axes if the accretion spin-up torque has a component that tends to align the star's magnetic field with its spin axis as it spins up. Motion of the magnetic poles toward the spin axis will be facilitated by diffusion of magnetic flux through the crust and/or fracture and fragmentation of the crust (Ruderman 1991).

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. If instead both poles are in the same rotation hemisphere when spin-up begins, they will be dragged toward the same spin pole. In either case, the strength of the magnetic field at the stellar surface will be orders of magnitude larger than the strength of its dipole component, as we now explain.

An initial magnetic field M₁ that is approximately uniform over a stellar hemisphere of radius R has a magnetic moment μ₁ ≈ M₁R³. The magnetic flux threading the fluid core is conserved as the magnetic field is squeezed. Suppose the inward moving vortices squeeze the flux into a tube of radius a ≈ R/30. This will produce a squeezed magnetic field of strength M₂ ≈ (R/a)²M₁ ≈ 10³M₁.

If the star's north and south magnetic poles are forced toward opposite spin poles, the dipole moment of the squeezed magnetic field will be aligned with the star's spin axis and will have a magnitude μ₂ ≈ M₂Ra² ≈ μ₁, corresponding to a surface magnetic field with a dipole component M₂(dipole) ≈ M₁. The full strength M₂ of the surface magnetic field will therefore be ≈(R/a)² ≈ 10³ times larger than the strength M₂(dipole) of its dipole component.

If instead the star's north and south magnetic poles are both forced toward the same spin pole, the dipole moment of the squeezed magnetic field will be orthogonal to the spin axis and hence the star will be an "orthogonal rotator," even though both magnetic poles are near a spin pole rather than near the spin equator. In this case the dipole moment will have a magnitude μ₂ ≈ M₂La² ≈ (L/R)μ₁, where L is the final separation between the north and south magnetic poles, corresponding to a surface magnetic field with a dipole component M₂(dipole) ≈ (L/R)M₁. Thus if L ≈ 0.1R, the dipole moment of the squeezed field will be about 10 times smaller than the dipole moment of the original field and the full strength M₂ of the surface magnetic field will be ≈(R/L)(R/a)² ≈ 10⁴ times larger than the strength M₂(dipole) of its dipole component.

Regardless of whether the star's north and south magnetic poles are forced toward opposite spin poles, creating an aligned rotator, or toward the same spin pole, creating an orthogonal rotator, both magnetic poles will end up very close to the spin axis. Accreting gas that is channeled along magnetic field lines will therefore impact the stellar surface close to the spin axis, and accretion-powered X-ray emission will come primarily from regions near the spin axis.

Note that if a pulsar's north and south magnetic poles have both been forced close to the same spin axis, the flow of accreting matter could impact the surface in a pattern with predominantly either onefold or twofold symmetry relative to the spin axis. In this accretion geometry, even a small change in the properties of the flow through the inner disk could change the dominant symmetry of the emitting area from onefold to twofold or vice versa, thereby changing the relative strengths of the first and second harmonics in the pulse profile.

Another consequence of this picture of magnetic field evolution is that both accretion-powered (X-ray) and rotation-powered (radio) millisecond pulsars may have surface magnetic fields ∼10¹¹–10¹² G. This is ∼10³–10⁴ times stronger than the ∼10⁸–10⁹ G dipole components inferred from the magnetospheric radii of AMXPs (see, e.g., Psaltis & Chakrabarty 1999; Lamb & Boutloukos 2008), the spin-down of SAX J1808.4−3658 in quiescence (Hartman et al. 2008), and the spin-down rates of the rotation-powered millisecond radio pulsars (see Lamb & Boutloukos 2008). Magnetic fields this strong are strong enough to create an accumulation point for fuel for thermonuclear bursts and produce strong-magnetic-field features in the keV X-ray spectra of AMXPs (see Section 4.4).

The picture of the X-ray emission and magnetic field evolution of AMXPs that we 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.

If this explanation of the transient nature of the AMXPs is correct, they should be spinning down on long timescales, as seems to be the case (see, e.g., Hartman et al. 2008), and the amplitudes and phases the harmonic components of their waveforms should evolve in a correlated way as their magnetic poles move away from their spin axes (see Section 4).

Rotation-powered millisecond radio pulsars (MRPs) are thought to be the offspring of AMXPs and therefore should have magnetic field geometries similar to those of the AMXPs, except that the low rate of accretion at the end of the accretion phase and magnetic dipole braking toward the end of the accretion phase and afterward will tend to spin them back down, forcing their magnetic poles away from their spin axes.

Although the radio emission properties of most MRPs provide little clear evidence about the locations of their magnetic poles (see Manchester & Han 2004), there are indications from their radio emission that the dipole moments of a substantial number of the most rapidly spinning MRPs in the galactic disk are nearly aligned or nearly orthogonal to their spin axes (Chen et al. 1998; see also Chen & Ruderman 1993). 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 X-ray 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). We 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. If so, their magnetic poles may have been forced away from their spin axes by a similar factor. This could allow consistency between the ∼40°–60° inclinations inferred by Bogdanov et al. for these three MRPs and the ≲20° inclinations typically required in the model of AMXPs described here. We also note that only MRPs that produce high-amplitude X-ray oscillations are currently detectable as oscillators, so the ones that are detected may have magnetic inclinations larger than is typical. More investigation is needed.

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 reason is that the properties of pulses are determined by the location of the emitting area, which depends on the rate and structure of the accretion flow through the inner disk. The X-ray luminosity and spectrum of the pulsar will also depend on the accretion flow through the inner disk. Hence the properties of pulses should be correlated with the X-ray luminosity and spectrum.

Changes in the location of the emitting area on the stellar surface on timescales longer than the ∼10²–10³ s intervals required to construct stable pulse profiles will produce correlated changes in the amplitudes, harmonic content, and arrival times of these profiles.

In the nearly aligned moving spot model, the arrival times of pulses with low amplitudes are expected to fluctuate much more than the arrival times of pulses with high amplitudes (see Figure 6). The reason is that pulse amplitudes are lower when the emitting area is closer to the spin axis, where small variations in the position of the emitting area can produce large variations in the pulse arrival time.

A second expectation is that pulse phase residuals will form a track in the phase-residual versus pulse-amplitude plane. This will be the case if the area where accreting matter impacts the stellar surface and X-rays are emitted tends to move along a favored path on the stellar surface as the accretion flow through the inner disk changes. 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. Changes in the accretion flow through the inner disk will therefore cause correlated changes in the phases and amplitudes of pulses, producing a track when these are plotted in the phase-residual versus pulse-amplitude plane. Such tracks are likely to be more evident if the range of the pulse-amplitude variation is large.

The pulses of IGR J00291+5934, XTE J1751−305, XTE J1807−294, and XTE J1814−338 seem to form track-like patterns in the phase-residual versus pulse-amplitude plane when the pulse amplitude is ≳5% (see Section 4.3). An important expectation of the moving spot model is that the position of pulses along such a track should be correlated with the X-ray luminosity and spectrum of the pulsar.

Another check of the moving spot model is possible if the nearly identical pulse profiles and phases of the accretion- and nuclear-powered X-ray oscillations of XTE J1814−338 and SAX J1808.4−3658 are produced by emission from the same area on the stellar surface, as suggested in Section 4.4. If this is the case, the model predicts that longer-term (days–weeks) variations of the phase residuals of the accretion- and nuclear-powered 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. Evidence of this behavior would support the moving spot model.

The location of the emitting area on the stellar surface is likely to change on timescales as short as the ∼1 ms dynamical timescale in the inner disk. These rapid movements of the emitting area will cause rapid variations of the observed X-ray flux. Flux variations on timescales shorter than the ∼10²–10³ s intervals required to construct stable pulse profiles are observable as noise in excess of the photon counting noise. 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. Hence the strength of the excess noise produced by motion of the emitting area should be correlated with these other properties of the AMXP.

A strong expectation in the nearly aligned moving spot model is that the component of the excess noise produced by motion of the emitting area should be stronger when the pulse amplitude is smaller and weaker when the pulse amplitude is larger (see Section 4.3).

One way to focus on the strength of the noise produced by movement of the emitting area would be to filter the X-ray flux data to remove variations on timescales shorter than, say, 0.05 s. This would remove the variability associated with oscillations at the pulsar spin frequency and any high-frequency QPOs. The remaining total variability would then be correlated with the other properties of the AMXP listed above. Detection of greater variability when the observed pulse amplitude is smaller would support the nearly aligned moving spot model.

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. Such features may include increased flux below the fundamental cyclotron resonance energy, where the magnetic field suppresses the opacity of surface layers (see, e.g., Pavlov et al. 1975); resonance scattering features produced by the fundamental cyclotron resonance and its overtones (see, e.g., Mészáros & Nagel 1985); and changes in the spectrum at the higher energies where the cyclotron scattering becomes unimportant.

Another implication of the existence of AMXPs with surface magnetic fields as strong as ∼10¹¹–10¹² G is that the rotation-powered millisecond pulsars that are their progeny should have surface magnetic fields this strong, even though their dipole fields are ∼10⁸–10⁹ G. Magnetic fields this strong can be expected to affect particle acceleration and γ-ray emission by these pulsars.