The Orbital Period of the Accreting Pulsar GX1+4

We report strong evidence for a ~304-day periodicity in the spin history of the accretion-powered pulsar GX1+4 that is most probably associated with the orbital period of the system. We have used data from the Burst and Transient Source Experiment on the Compton Gamma Ray Observatory to show a clear periodic modulation of the pulsar frequency from 1991 to date, in excellent agreement with the ephemeris proposed by Cutler, Dennis&Dolan (1986). Our results indicate that the orbital period of GX1+4 is 303.8 +- 1.1 days, making it the widest known low-mass X-ray binary system by more than one order of magnitude and putting this long-standing question to rest. A likely scenario for this system is an elliptical orbit in which the neutron star decreases its spin-down rate (or even exhibits a momentary spin-up behavior) at periastron passages due to the higher torque exerted by the accretion disk onto the magnetosphere of the neutron star. These results are not inconsistent with both the X-ray pulsed flux light curve measured by BATSE during the same epoch and the X-ray flux history from the All-Sky Monitor (ASM) onboard the Rossi X-Ray Timing Explorer.


Introduction
GX 1+4 is a bright Galactic Center accretion-powered pulsar in a low-mass x-ray binary system (LMXB) discovered in the early 1970s (Lewin, Ricker & McClintock 1971).
Throughout the 1970s the pulsar exhibited a spin-up behavior with the pulsation period decreasing from 135 s to less than 110 s (Cutler, Dennis & Dolan 1986 -hereafter CDD86 -and references therein), corresponding to a spin-up rate ofṖ ∼ −2 s/year. After experiencing an extended low-intensity state in the early 1980s (Hall & Develaar 1983;McClintock & Leventhal 1989), GX 1+4 re-emerged in a spin-down state (Makishima et al. 1988;Sakao et al. 1990) with approximately the same |Ṗ | and stayed in this state ever since, with occasional short-term variations ofṖ .
Infrared observations and optical spectroscopy of GX 1+4 established a rare association of a neutron star with a M5 III giant star, V2116 Oph, in a symbiotic binary system (Glass & Feast 1973;Davidsen, Malina & Bowyer 1977;Chakrabarty & Roche 1997). The identification was made secure by a ROSAT accurate position determination (Predehl, Friedrich & Staubert 1995) and by the discovery of optical pulsations in V2116 Oph consistent with the spin period of the neutron star , Pereira et al. 1997. In comparison with the other four known LMXB accretion-powered pulsars (GRO J1744−28, Her X-1, 4U 1627−67 and the recently discovered millisecond accreting pulsar SAX J1808.4-3658 - Wijnands & van der Klis 1998), GX 1+4 has a much longer (factor of ∼ 100) spin period and its orbital period, albeit not securely measured until this work, was known to be at least one order of magnitude longer than the periods of the other systems. Physically quantitative lower limits on the binary period of GX 1+4 were derived by Chakrabarty and Roche (1997), who showed that the binary period must be at least 100 d, and is probably more than 260 d.
In 1991, the Burst and Transient Source Experiment (BATSE) on the Compton Gamma Ray Observatory (CGRO) initiated a continuous and nearly uniform monitoring of GX 1+4. The BATSE observations confirmed the spin-down trend with occasional dramatic spin-up/down torque reversal events (Chakrabarty 1996, Nelson et al. 1997. Attempts to find the orbital period of GX 1+4 by Doppler shifts of the pulsar pulse timing or optical lines have both been inconclusive so far. For the X-ray timing measurements, the accretion torque magnitude is much larger than the expected orbital Doppler shifts, and the torque fluctuations have significant power at the time scales comparable to the expected binary period (Chakrabarty 1996). In the case of the optical lines, the problem is the long period ( > ∼ 100 days) expected (Davidsen, Malina & Bowyer 1977, Doty, Hoffman & Lewin 1981Sood et al. 1995). Long-term optical photometry in R band has shown variations in the light curve with periods of ∼ 30 and ∼ 110 days (Pereira, Braga & Jablonski 1996;Pereira 1998). Using a small number of X-ray measurements carried out during the spin-up phase of GX 1+4 in the 1970s, CDD86 produced an ephemeris for predicting periodical enhancements in the spin-up rate of the neutron star. A possible interpretation for this periodic behavior is that the neutron star and the red giant are in an elliptical orbit with a 304-day period.
In this work we report the results of a comprehensive time-series analysis of the BATSE data on GX 1+4 in an attempt to find the orbital period of the system. We discuss the implications that can be drawn from our results in light of the possible models for this source and show that the elliptical orbit interpretation is probably the correct one. We present a refined version of the ephemeris originally proposed by CDD86.

Data Analysis and Results
The frequency and the pulsed flux data between Julian Day (JD) 2448376.5 and 2451138.5 (i.e., 1991April 29 to 1998 used in this work were obtained from Chakrabarty (1996) and from the BATSE public domain data available at i.e., no monocromatic or even QPO signals are produced. By comparing the amplitude of our 302-day peak with the local value obtained by the mean of the numerical simulations (the peak is a factor of 13.91 higher), we obtain a statistical significance of 99.98% for the detection. Epoch folding the data using the 302-day period yields a 1-σ uncertainty of 1.7 days.
In the pulsed flux data, the most interesting feature is a wide structure of lowsignificance peaks observed in the range 200-500 days, with no significant peak at ∼ 300 days.
By analyzing the variation of the period of GX 1+4 during the spin-up phase in the 1970s, CDD86 proposed a 304 -day orbital period and an ephemeris to predict the events of enhanced spin-up: T = JD 2, 444, 574.5 ± 304 n, where n is an integer. This ephemeris is based on four events discussed by the authors, whose existence was inferred from ad-hoc assumptions and extrapolations of the observations. The projected enhanced spin-up events derived from that ephemeris for the epochs contained in the BATSE dataset, represented as solid vertical lines in the lower panel of Fig. 1, are in excellent agreement with the BATSE reduced spin-down and spin-up events. The BATSE dataset is obviously significantly more reliable than CDD86's inasmuch as it is based on 9 well-covered events measured with the same instrument as opposed to the 4 events discussed in CDD86. The striking agreement of CDD86's ephemeris with the BATSE observations is very conspicuous and give a very strong support to the claim that the orbital period of the system is indeed ∼ 304 days.
Taking integer cycle numbers, with the T 0 epoch of CDD86 as cycle −23, and performing a linear least-squares fit to the frequency residuals seen in the lower panel of Fig. 1, we find that the following ephemeris can represent the time of occurrence T of the maxima in the frequency residuals: where n is any integer. The events predicted by the above ephemeris are shown as vertical dashed lines in the three panels of Fig. 1. Taking into account a conservative uncertainty estimate of 30 days for the peaks of the BATSE events, the reduced χ 2 of the fit is χ 2 r = 0.61. The value of 303.8 ± 1.1 days for the orbital period is consistent with the one obtained through power spectrum analysis performed on the BATSE data, which gives further support for the period determination.

Discussion
In the BATSE era, the long term frequency history of GX 1+4 shown in Fig. 1  Frequency derivative reversals occur on times preceding the epochs of events labeled # 5, 7 and 9 in the bottom panel. The upper panel of Fig. 1 shows that these events are somewhat correlated with rather intense flares in the pulsed flux. In the 1970s, when the measurements used by CDD86 were carried out, the source was in a spin-up extended state.
The scenario proposed by CDD86 to explain the periodic occurrence of enhanced spin-up events was that the system was in a elliptical orbit and the periastron passages would occur whenṖ is maximum, as expected in standard accretion from a spherically expanding stellar wind. It is widely accepted today, as inferred from GX 1+4's optical/IR properties Chakrabarty, van Kerkwijk & Larkin 1998;Chakrabarty et al. 1997;Chakrabarty & Roche 1997), that the system has an accretion disk.
Since the neutron star is currently spinning-down, the radius at which the magnetosphere boundary would corotate with the disk, r co = (GMP 2 /4π 2 ) 1/3 ∼ 3.6 × 10 4 P 2/3 100s km, where M is the mass of the neutron star (assumed to be ≈ 1.4 M ⊙ ) and P 100s is the spin period in units of 100 seconds, is probably smaller than the magnetosphere radius r M ∼ 4.1 × 10 4 L −2/7 36 km, where L 36 is the X-ray luminosity in units of 10 36 erg/s (Frank, King & Raine 1992). This value for r M assumes a surface magnetic field of ∼ 10 14 G for GX 1+4 (Makishima et al. 1988, White 1988, Cui 1997. Since the pulse period is ∼ 120 s and the luminosity is typically < ∼ 10 37 erg/s, the period is close to the equilibrium value, for which r co ∼ r M . This allows spin-down to occur even though accretion continues, the centrifugal barrier not being sufficiently effective (White 1988).
Assuming that the elliptical orbit is the correct interpretation for the origin of the modulation, the mass accretion rate (and hence the luminosity) should increase as the neutron star approaches periastron, making r M approach r co . As the velocity gradient between the disk material and the material flowing along the magnetic field lines decreases, the spin-down torque gets smaller and the neutron star decelerates at a slower rate.
We expect that this mechanism will produce a peak in the frequency residuals close to the periastron epoch, beyond which the neutron star will start to get back to a higher spin-down rate. Occasionally, due to the highly variable mass loss rate of the red giant, r co will surpass r M and the neutron star will spin-up for a brief period of time during periastron, as observed in the BATSE frequency curve in events 5, 7 and 9. According to this picture, one would expect an increase in X-ray luminosity at periastron. Although this is only marginally indicated in the BATSE pulsed flux light curve, it should be pointed out that total flux data from the All Sky Monitor (ASM) onboard RXTE for the epoch MJD 50088 to 51044 does not correlate significantly with the BATSE pulsed flux, indicating that the pulsed flux may not be a good tracer of the accretion luminosity in this system.
Furthermore, the periodic ∼ 5µHz excursions in the residual frequency would lead to very low-significance variations in the X-ray flux measured by the ASM, as we now show. Taking the fiducial torque N 0 =Ṁ √ GM X r co given by Bildsten et al. (1997), whereṀ is the accretion rate and M X is the mass of the neutron star, as an order-of-magnitude estimation (since r co ≈ r M ), we can establish a lower limit to the variation inṀ (∆Ṁ ) that produced the residual torque, using the fact thatν = N 0 /2πI, where I is the moment of inertia of the neutron star (Ravenhall & Pethick 1994). Since the relative variation in flux (F ) scales as the relative variation in luminosity, we get ∆F/F ∼ 0.3 for L ∼ 10 37 erg/s. The typical ASM GX 1+4 flux is ∼ 1 ± 2 count/s in the 2-10 keV, so the expected variations of ∼ 0.3 counts/s would be very hard to detect, given the available observational coverage. This is consistent with the lack of any significant periodic signal in our calculation of the power spectrum of the entire available ASM flux history of GX 1+4 (from MJD 50088 to 51353).
In the elliptical orbit interpretation, one would also expect that tidal torques exerted by the red giant envelope would circularize the orbit in a short time scale (Verbunt & Phinney 1995). However, with a period of ∼ 300 days, the red giant radius is probably less than 7% of the binary separation, as shown below. Since the rate scales as (R c /a) −8 , where R c is the red giant radius and a is the binary separation, we do not expect the circularization time scale to be too short. Furthermore, Verbunt & Phinney (1995) show that for orbital periods longer than about 200 days, the eccentricities of red giant binaries in open clusters span the full range.
An alternative interpretation for the observed modulation would be the presence of oscillation modes in the red giant star. For an M5 giant, persistent radial oscillations with a period of ∼ 300 days are perfectly plausible (Whitelock 1987). In this case, the oscillations could excite a modulation in the mass loss rate through the stellar wind that could produce the modulated torque history. However, the stability of the infrared magnitudes of V2116 Oph (Chakrabarty & Roche 1997) preclude it from being a long-period variable, since these stars undergo regular > ∼ 1 mag variations in the infrared (Whitelock 1987). In addition, the secular optical light curve in the R band obtained by our group at Laboratório Nacional de Astrofísica (Brazil) from 1991 to date shows no signs of these oscillations (Pereira, Braga & Jablonski 1996;Pereira 1998).
It is noteworthy that the amplitude of the residual frequency oscillations in GX 1+4 cannot be attributed to Doppler shifts (Chakrabarty 1996). A firm lower limit for the companion mass is given by the X-ray mass function f X (M) = (c∆ν/ν) 3 P orb /2πG, which would be equal to ∼ 210 M ⊙ for a ∼ 5µHz amplitude and a 304-day orbital period. This is clearly too massive for a red giant and actually for any stellar companion. There is also no evidence of Doppler shifts in the spectral lines of V2116 Oph (Sood et al. 1995;Chakrabarty & Roche 1997), which could be an indication that the inclination of the system is fairly low.
The spectral and luminosity classification of V2116 Oph, together with the measured interstellar extinction of A V ≈ 5, is consistent with a low-mass star (M ∼ 0.8−2 M ⊙ ) on the first-ascent red giant branch at a distance of 3−6 kpc (Chakrabarty & Roche 1997). The range of radii for such a start is ∼ 50 − 110 R⊙. The size of the Roche lobe of this object as the companion in the binary system can be estimated by the radius of a sphere with the same volume as the lobe, where q = M g /M X is the mass ratio of the red giant and the neutron star, M X is in solar mass units and P d is the orbital period in days (Eggleton 1983  , we get P orb ∼ 270 ± 82 days for L X ∼ L Edd , which is fully consistent with our results. It should be noted, however, that this model is based upon the assumption that the optical emission is dominated by reprocessing of X-rays in the accretion disk, which is not clear to be the case in GX 1+4.
In conclusion, we have shown that the long-sought orbital period of GX 1+4 is very likely to be 304 days, as proposed in 1986 by CDD86 with marginal confidence. A more thorough covering of the X-ray luminosity of the system, with high sensitivity and spanning several cycles, will be very important to test the elliptical orbit model.