THE ASTROPHYSICAL JOURNAL, 490:L87–L90, 1997 November 20
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Millisecond X-Ray Pulsars in Low-mass X-Ray Binaries

NICHOLAS E. WHITE 1 AND WILLIAM ZHANG 1

Laboratory for High Energy Astrophysics, Goddard Space Flight Center, Greenbelt, MD 20771

Received 1997 August 5; accepted 1997 September 24; published 1997 October 23


ABSTRACT

     The discovery of millisecond X-ray pulsars in low-mass X-ray binaries (LMXRBs) with the Rossi X-Ray Timing Explorer (RXTE) has resulted in the determination of the neutron star spin periods to be in a very narrow period range. Based on evolutionary models for LMXRBs, it is likely that these pulsars have accreted sufficient material to be at or close to their equilibrium spin periods. If this is the case, then the similar neutron star rotation periods over 2 orders of magnitudes in observed luminosity require a comparable magnetospheric radius in all these systems to give a similar spin period. This means that either there is a correlation between surface magnetic field strength and X-ray luminosity (B∝L$\mathstrut{^{1{/}2}_{{\rm X}}}$ for the commonly used magnetospheric scaling for a gas pressure dominated accretion disk) or, if the neutron star surface magnetic field is comparable in all cases, the magnetospheric radius is very weakly dependent on the accretion rate in the inner radiation-dominated disk relevant to this regime. We suggest that the anomalous case of Sco X-1, where there is an apparent change in the inferred neutron star spin period, may be understood in terms of the radiation-driven expansion of the neutron star photosphere by up to 30%.

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

CONTENTS

FOOTNOTES

     1 Code 662, NASA/GSFC Greenbelt, MD 20771.

§1. INTRODUCTION

     Millisecond radio pulsars (MRPs) are thought to be old neutron stars recycled to rapid rotation by accretion in LMXRBs (see Bhattacharya 1995, and references therein). Up until recently observations of the critical neutron star parameters (mass, rotation period, and magnetic field strength) during its recycling phase as a bright X-ray binary source had proved elusive. This situation has changed dramatically with the launch of NASA's Rossi X-Ray Timing Explorer (RXTE), which has provided the first direct X-ray signature of millisecond pulsars in LMXRBs. The millisecond rotation period of the neutron stars in five LMXRBs (4U 1728-34, 4U 1636-53, KS 1731-26, X1743-29, and Aql X-1) is seen directly as a near coherent X-ray pulsation during X-ray bursts (Strohmayer et al. 1996; Zhang et al. 1997; Smith, Morgan, & Bradt 1997; Strohmayer et al. 1997; Zhang et al. 1998). Pulsations due to the neutron star rotation are usually not seen away from the X-ray bursts but can still be reliably determined using two simultaneously present quasi-periodic oscillations (QPOs) seen between 325 and 1193 Hz. While the centroid frequency of the two peaks can vary, correlated with source intensity, in seven out of eight sources the two peaks maintain a constant frequency separation of 263–330 Hz (Strohmayer et al. 1996; Smale, Zhang, & White 1997; Wijnands & van der Klis 1997; Wijnands et al. 1997; Ford et al. 1997; van der Klis et al. 1996b, 1997a). The most likely origin of the lower QPO frequency is a beat between a characteristic frequency in the inner accretion disk (the higher QPO frequency) and the neutron star spin. The crucial evidence that supports this conclusion came from observations of 4U 1728-34 by Strohmayer et al. (1996), who detected a millisecond pulsation during several X-ray bursts at exactly the QPO difference frequency. There are now two other similar cases in which the frequency detected during the burst matches the QPO difference frequency (Zhang et al. 1997; Wijnands & van der Klis 1997; Smith et al. 1997). Corroborating evidence (at 4 σ significance) for the beat frequency model comes from a third peak seen on one occasion in the quiescent flux of 4U 0614+09 at the QPO difference frequency (Ford et al. 1997).

     In this Letter we start from the assumption that with RXTE we are for the first time determining the spin period of the underlying millisecond X-ray pulsars (MXPs) in LMXRBs. We first suggest an explanation for the exceptional case of Sco X-1 where the frequency separation of the QPO peaks is not constant but rather decreases with increasing accretion rate, which has raised a concern regarding the reliability of the double-peaked QPO as an indicator of the underlying neutron star spin (van der Klis et al. 1997b). We use the measured spin periods and observed luminosities to investigate the magnetosphere-disk interaction in these systems.

§2. THE SAMPLE

     Table 1 summarizes the currently known millisecond X-ray pulsar periods and how the period was determined. Three (4U 1728-34, KS 1731-26, and 4U 1636-63) are from both the QPO difference frequency and pulsations seen during the X-ray burst. In two (KS 1731-26 and 4U 1636-63) out of these three cases the pulsation seen during a burst is at half the period inferred from the QPO frequency difference, suggesting a double peak pulsation during the burst (Zhang et al. 1997; Wijnands & van der Klis 1997). Such a double-peaked pulse is commonly seen in longer period X-ray pulsars and arises because of viewing two poles. The pulsation amplitude of the millisecond pulsations during the burst is 5%–10% and may arise from asymmetries in the nuclear burning on the neutron star surface, perhaps related to the two magnetic poles. Another five period determinations use just the difference frequency, and two others (Aql X-1 and X1743-28) come from pulsations seen during X-ray bursts. The 10 pulsar periods determined to date show a bimodal period distribution, clustering around either 1.7–1.9 or 2.7–3.8 ms. The sources with periods between 1.7 and 1.9 ms are also the ones in which the pulsation period has only been observed during an X-ray burst. If this represents a second harmonic of the true rotation period, then the spin period distribution would be single-peaked and cover a narrow range of 2.7–3.8 ms. On the other hand, if the burst periods represent the true period, and the period obtained from the QPO peak separation is in most cases twice the true value, then the period distribution will cover 1.7–2.7 ms. For simplicity we will assume the QPO difference frequency represents the true underlying spin period, although the conclusions will not change if the other possibility turns out to be the case.

     The frequency separation of the QPO twin peaks seen in Sco X-1 does not remain constant but rather varies from 230 to 310 Hz with the highest frequency occurring at the lowest luminosity (van der Klis et al. 1997b). Sco X-1 has not shown X-ray bursts, so there has not been an independent determination of the neutron star spin. At the start of Eddington-limited X-ray bursts the pulse period is seen to decrease in the same sense by a few percent and then quickly settles back to a stable value (Strohmayer et al. 1997; Zhang et al. 1998). This period change is most likely caused by the expansion of the neutron star photosphere by 20–50 m, with conservation of angular momentum causing the rotation of the atmosphere to slow as it expands (Strohmayer et al. 1997). Sco X-1 is thought to be accreting close to the Eddington limit for a neutron star (Hasinger & van der Klis 1989). Just as is the case for the Eddington-limited X-ray bursts, an increase in accretion rate increases the radiation pressure and may cause the photosphere to lift off.

     A spherical shell on the neutron star surface that participates in the expansion having a mass of M, an inner radius of r, and an outer radius of r+Δr, has a moment of inertia



where x = Δr/r. Prior to any expansion, i.e., Δr=0, the moment of inertia of the shell is I$\mathstrut{_{0}}$=$\mathstrut{\frac {2}{3} }$Mr$\mathstrut{^{2}}$. To compensate for the 35% (3.23–4.35 ms) decrease in spin rate, the moment of inertia of the shell has to increase by 35%, which according to the above equation will amount to a ∼30% expansion of the neutron star photosphere (∼3 km for a 10 km neutron star). This is 60 times larger than inferred from the frequency shift seen during bursts. But it is well within the radius expansion events up to 100 km determined from the blackbody spectra during X-ray bursts (see, e.g., Haberl & Titarchuk 1995). The longest period occurs when the mass accretion rate is highest, in accord with a radiation-driven expanding atmosphere model. We also note that as predicted by this model the area of the blackbody component detected in the spectrum of Sco X-1 increases with increasing accretion rate (White, Peacock, & Taylor 1985; van der Klis et al. 1987).

     Our model is consistent with a sonic point beat frequency model in which the inner disk radius is driven by the conditions in the inner accretion disk and does not involve threading inflowing material onto the magnetic field lines (Miller, Lamb, & Psaltis 1998). It is not consistent with the Alpar & Shaham (1985) magnetic gating model in which blobs of material enter the magnetosphere at the beat between the Keplerian disk and magnetosphere rotation periods. In this case the magnetic field would remain tied to the underlying neutron star as the photosphere expands and should retain the underlying neutron star rotation period. In Table 1 we use for Sco X-1 the shortest period of 3.23 ms as representing the rotation period of the neutron star.

§3. MAGNETIC FIELD VERSUS SPIN PERIOD

     The angular momentum added to the neutron star as a result of the mass transfer will accelerate its spin. The neutron star reaches spin equilibrium when the rotation period equals the Keplerian period of the inner accretion disk at the magnetospheric boundary. The timescale to spin a neutron star up to its equilibrium period depends both on the mass accretion rate history and the magnetic field decay of the neutron star. If the magnetic field decay timescale is longer than the spin-up timescale, then a neutron star with a constant magnetic field of 108 G at an accretion rate of 10-8 M⊙ yr-1 will be spun up to equilibrium in ∼107 yr (see Urpin & Konenkov 1997). The various evolutionary tracks that have been identified for LMXRBs suggest that the neutron star should be spun up to either its equilibrium period, or maximum rotation period of 0.5–1.0 ms within 107 yr (see, e.g., White, Stella, & Parmar 1989; Burderi, King, & Winn 1996). The old disk population galactic distribution of LMXRBs indicates they are very old systems (van Paradijs & White 1995). One possible way around this argument is that the more luminous systems have a lower time-averaged accretion rate that dictates the equilibrium period, and we are observing them in relatively short lived outburst. While we cannot rule this out completely, the fact that all the high-luminosity LMXRBs have been active at a similar level for the past 30 yr makes it unlikely.

     For a given pulse period and mass accretion rate, the assumption of spin equilibrium sets a maximum surface magnetic field strength B given by



where Mns/1.4 M⊙ is the neutron star mass in units of 1.4 M⊙, Rns/10 km is the neutron star radius in units of 10 km, P$\mathstrut{_{2\,{\rm ms}}}$ is the neutron star spin period in units of 2 ms, $\mathstrut{{\ucpmathaccent{M}{"705F}}}$$\mathstrut{_{-8}}$ is the mass accretion rate in units of 10-8 M⊙ yr-1, and ωc is the critical fastness parameter, which is used to correct for reverse torques in the complex disk-magnetosphere boundary region acting to spin down the neutron star as it approaches spin equilibrium. The value of ωc is not well constrained but is thought to lie between 0.35 and 0.9 (Ghosh & Lamb 1979, 1991). From a study of the spin and orbital periods of MRPs in binaries Burderi et al. (1996) find a lower value of ωc of ∼0.1. Decreasing ωc will decrease the inferred magnetic field strength for a given mass accretion rate (eq. [2]) or increase the equilibrium spin period for a given set of neutron star and accretion parameters. Equation (2) assumes that the magnetospheric radius is determined by a gas pressure–dominated accretion disk (a spherical flow gives a similar scaling). We will discuss later that this assumption may not be appropriate for the inner radiation-dominated disk zone relevant here.

     By converting the observed luminosity L$\mathstrut{_{{\rm X}}}$ into an accretion rate using L$\mathstrut{_{{\rm X}}}$=1.2×10$\mathstrut{^{38}}$$\mathstrut{{\ucpmathaccent{M}{"705F}}}$$\mathstrut{_{-8}}$ ergs s$\mathstrut{^{-1}}$, equation (2) gives an upper limit to the surface field of the neutron star for each pulsar. We have taken the best estimate of the distances available from van Paradijs & White (1995). The fluxes are taken from van Paradijs (1995) and represent the average values for each source. For the transient source Aql X-1 we take an average outburst value of ∼100 μJy. For those cases where it is unclear if the frequency detected represents the first or second harmonic, we have taken the longer period since this gives the maximum B. In Table 1 we summarize the parameters used and the derived upper limit on B. The maximum B implied by equation (2) ranges from 2×10$\mathstrut{^{8}}$ to 2×10$\mathstrut{^{9}}$ G (using the maximum value of ω$\mathstrut{_{c}}$=1).

     In Figure 1 we plot B versus spin period, along with the spin equilibrium lines for three accretion rates of 10-10, 10-9, and 10-8 M⊙ yr-1. The ambiguity in the neutron star spin period for the four cases in which X-ray bursts are seen causes the points to slide downward along the lines of constant accretion rate (Fig. 1). There are three distinct groups of sources evident in Figure 1. The tightest constraint on B comes from the four lowest luminosity systems with ∼1036 ergs s-1 Aql X-1, X1743-29, 4U 0614+09, and KS 1731-26, where B≤2×10$\mathstrut{^{8}}$ G. There is an intermediate luminosity group with ∼1036 ergs s-1, where B≤6×10$\mathstrut{^{8}}$ G, and an Eddington-limited group (the Z sources), where B≤2×10$\mathstrut{^{9}}$ G. To better illustrate the luminosity dependence of the upper limits on B, in Figure 2 we show the B versus X-ray luminosity relationship. The upper limits follow B∝L$\mathstrut{^{1{/}2}_{{\rm X}}}$ (solid line). The four cases in which the neutron star spin period might be half the given value are also shown. These four points also lie along a B∝L$\mathstrut{^{1{/}2}_{{\rm X}}}$ line. If a mix of the pulsars have half the period and others do not, then taken as a whole this will give a steeper relationship between B and L$\mathstrut{_{{\rm X}}}$.

Fig. 1 Fig. 2

§4. DISCUSSION

     The new capability of RXTE that combines large collecting area with submillisecond timing has allowed for the first time the determination of the spin periods of the long-suspected millisecond pulsars in LMXRBs. It is remarkable that the spin periods for all MXPs irrespective of X-ray luminosity span a narrow range of period between 2.7 and 3.8 ms (or 1.7 and 2.7 ms if the periodicity detected during bursts represents the first harmonic). Either range is consistent with the short end of the period distribution of MRPs in the Parkes southern pulsar survey, which shows a broad distribution from 2 to 10 ms and a peak at 4–6 ms (Lyne et al. 1997). If all the MXP systems are at or close to spin equilibrium, then the narrow range of rotation period requires a similar magnetospheric radius for a given neutron star mass, representing the inner Keplerian orbit of the accretion disk. We have found that if the magnetospheric radius scales as expected for a gas pressure–dominated accretion disk (or spherical accretion), then the 2 orders of magnitude range in X-ray luminosity requires B∝L$\mathstrut{^{1{/}2}_{{\rm X}}}$ (Fig. 2). This exactly reflects the dependence in equation (2) used to derive the upper limits to the B field and comes from the need to maintain a constant magnetospheric radius over 2 orders of magnitude in accretion rate. This result depends both on the assumption that the pulsars are in spin equilibrium and that we are using the correct scaling law for the magnetospheric radius with accretion rate. We have argued in § 3 that it is very probable that these pulsars are at or close to spin equilibrium.

     The LMXRBs have been put into two subclasses called Z and Atoll based on their timing and spectral properties by Hasinger & van der Klis (1989). They find that the Z sources Sco X-1 and Cyg X-2 overlap the evolving giant subclass of LMXRBs and that the Atoll sources are part of a subclass with shorter orbital periods driven by angular momentum loss (see Burderi et al. 1996, and references therein). Hasinger & van der Klis (1989) suggest that the difference in the X-ray properties of the two subclasses are principally due to higher neutron star magnetic fields in the Z sources of 109 G, compared to 108 G in the Atoll sources. We have indicated the subclass of each MXP in Table 1. The location of the Z sources in the B-P plane (Fig. 1) is consistent with them having higher B. This might be consistent with accretion-induced magnetic field decay models, which predict the resultant magnetic field to be less for higher accretion rates, because the crust field is more quickly frozen into the core (Urpin & Geppert 1995; Konar & Bhattacharya 1997). The systems with the highest accretion rates are those with the longer orbital periods (>100 days), and these do indeed have higher B fields than the shorter period systems (van den Heuvel & Bitzaraki 1995). But, in spite of this, there has been no expectation of a strong correlation between magnetic field strength and instantaneous accretion rate across the entire class of LMXRBs, including systems like Sco X-1, which has a one day orbital period.

     White & Stella (1987) point out that the magnetospheric radius dependence on the accretion rate in the inner radiation-dominated accretion disk (relevant to this regime) changes from that used for the outer gas pressure–dominated zone. The scaling law in the inner disk zone depends on the disk viscosity prescription, which is uncertain. Various possibilities have been considered and many give a magnetospheric radius that is less sensitive to mass accretion rate than in the gas pressure zone (White & Stella 1987; Ghosh & Lamb 1991). There remains a strong dependence of the magnetospheric radius on surface magnetic field strength, so that the final equilibrium period will depend on this. If the neutron star magnetic field is similar in all cases (a few 108 G), then this requires a very weak dependence between accretion rate and the radius of the magnetosphere in the radiation-dominated disk.

     We are left with two possible conclusions: (1) The instantaneous accretion rate and surface magnetic field strength are correlated, by coincidence opposite to the dependence of the magnetosphere radius on accretion rate, or (2) the neutron star surface magnetic field is comparable in all cases, and there is a very weak dependence between accretion rate and the radius of the magnetosphere in the radiation-dominated zone. The latter seems less contrived and offers a more natural explanation for the similar characteristics of the pulse periods and high-frequency QPOs from these LMXRBs over 2 orders of magnitude in luminosity.

ACKNOWLEDGMENTS

     We thank Tod Strohmayer, Pranab Ghosh, and Michiel van der Klis for advice and comments.

REFERENCES

FIGURES


Full image (79kb) | Discussion in text

     FIG. 1.—Upper limits to the magnetic field, B, vs. the rotation period for the MXPs given in Table 1. The arrows indicate the magnetic field derived assuming the longer pulse periods given in Table 1. The crosses are the periods found during the four bursts that have a period half the value given by the QPO difference period. The diagonal lines indicate the expected spin equilibrium lines for three different accretion rates of 10-10, 10-9, and 10-8 M⊙ yr-1. A critical fastness parameter of unity has been assumed.


Full image (52kb) | Discussion in text

     FIG. 2.—Magnetic field, B, vs. the X-ray source luminosity for the MXPs in Table 1. The symbols are the same as used in Fig. 1. The diagonal line illustrates a B∝L$\mathstrut{^{0.5}}$ relationship.

TABLES

TABLE 1
MILLISECOND X-RAY PULSAR PARAMETERS
SourceTypePeriod
(ms)		Method aLX
(1036 ergs s-1)		Max B
(108 G)		References
4U 1728-34...A2.76B, D10.84.51
4U 1636-53...A3.45 (or 1.72)B/2, D9.35.5 (or 2.5)2
Aql X-1...A3.64 (or 1.82)B1.22.1 (or 0.93)3
X1743-29...A3.40 (or 1.70)B1.01.76 (or 0.78)4
4U 0614+09...A3.05D1.31.75
Sco X-1...Z3.23D200.023.56
KS 1731-26...A3.8 (or 1.9)B/2, D1.32.3 (or 1.0)7
GX 5-1...Z3.03D200.021.88
GX 17+2...Z3.27D200.023.839
4U 1820-30...A3.63D10.66.1910

     a Method: B-period derived from pulsations observed in type I bursts; D-period derived from the frequency difference of the two QPOs.
     REFERENCES.— (1) Strohmayer et al. 1996; (2) Zhang et al. 1997, Wijnands et al. 1997; (3) Zhang et al. 1998; (4) Strohmayer et al. 1997; (5) Ford et al. 1997; (6) van der Klis et al. 1996a; (7) Smith et al. 1997, Wijnands & van der Klis 1997; (8) van der Klis et al. 1996b; (9) van der Klis et al. 1997a; (10) Smale et al. 1997.

Image of typeset table | Discussion in text
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