A SHAPIRO DELAY DETECTION IN THE BINARY SYSTEM HOSTING THE MILLISECOND PULSAR PSR J1910−5959A

PSR J1910−5959A (henceforth PSRA) is a millisecond pulsar (MSP) discovered in the Parkes Globular Cluster Pulsar Survey (D'Amico et al. 2001, hereafter Paper I). It has a spin period of 3.27 ms, and it orbits around a low-mass companion with M_C ⩾ 0.2 M_☉, assuming a pulsar mass M_P = 1.4 M_☉, with an orbital period P_B = 0.84 days. Because of its unusual position with respect to the core of the globular cluster NGC 6752 (its angular distance from the cluster center is θ_⊥ = 64, corresponding to 3.3 half-mass radii; Paper I), it has been questioned whether this object is a member of the cluster (Bassa et al. 2006). In fact, of all cluster-associated MSPs, PSRA has the largest angular distance from the core of the hosting cluster.⁹

The binary companion of PSRA has been recognized as a helium white dwarf (HeWD) star by Ferraro et al. (2003) and independently by Bassa et al. (2003), via observations taken with the ESO Very Large Telescope and the Hubble Space Telescope (HST). The values obtained by these authors for the mass of the binary companion of PSRA are in agreement (M_co ≃ 0.17–0.20 M_☉). Both authors also agree in concluding that the photometric properties of this star are consistent with the hypothesis that PSRA is associated with the globular cluster NGC 6752.

The radial velocity curve of the companion of PSRA has been investigated by Cocozza et al. (2006) and Bassa et al. (2006) through phase-resolved spectroscopy in the optical band with HST observations. The resulting values for the mass ratio of the binary system for the mass of the optical companion and the orbital inclination were also compatible with each other, although Bassa et al. (2006) obtained more stringent constraints (see Section 3.2 for details). Bassa et al. (2006) carefully discussed the association of this binary system with NGC 6752 and indicated a preference for non-association, though their arguments are not conclusive because of the large uncertainties on the parameters that would allow one to discriminate between the two scenarios (see Section 3.2 for details).

A refined ephemeris for PSRA was presented by Corongiu et al. (2006, hereafter Paper II), where a low eccentricity model was applied to describe the pulsar's orbit. The measured eccentricity was e ∼ 3 × 10⁻⁶ at a longitude of periastron ω ∼ 90°. Paper II discussed the possibility that such eccentricity was indeed a misinterpreted Shapiro delay. Nevertheless, Paper II maintained a conservative approach to this problem and deferred the revision of this issue until a significantly longer time span of timing data became available.

Correlations between the orbital period, the pulsar mass, and the companion mass have been obtained by Tauris & Savonije (1999, hereafter TS99) from their numerical calculations on the non-conservative evolution of close binary systems with low-mass donor stars (M_D < 2 M_☉), a 1.3 M_☉ accreting neutron star, and an orbital period P_B ⩾ 2 days at the beginning of their low-mass X-ray binary (LMXB) phase. These systems have diverging orbital separation and are the progenitors of low-mass binary millisecond pulsars (Bhattacharya & van den Heuvel 1991). TS99 obtained a positive correlation between the orbital period and the companion mass and a negative correlation between the orbital period and the pulsar mass. The predicted M_C–P_B correlation has been observed by van Kerkwijk et al. (2005) and Lorimer (2008). TS99 remarked on the discrepancy between the predicted M_P–P_B correlation and the available data. However, in most cases the determinations of the M_C and M_P were based on models for the companion atmosphere or on statistical hypothesis for the orbital inclination, often leading to large uncertainties. The observation of a Shapiro delay allows one to obtain a direct measurement of M_C and M_P and gives a strong test for the aforementioned correlations.

In this paper, we present timing data for PSRA collected over 12 years at the Parkes radio telescope. This data span is twice than that used in Paper II and allowed us to get better constraints on the rotational, positional, and orbital parameters for PSRA, and to detect the Shapiro delay in this system. The paper is organized as follows. In Section 2, we briefly describe the observing methods and present the timing solution based on the available data span; in Section 3, we describe the method for measuring the Shapiro delay, present our determination of the companion mass, and discuss our results and their implications. In Section 4, we summarize the measurements for the masses of MSPs and their HeWD companions, focusing on those binaries for which M_C and M_P have been obtained from the detection of the Shapiro delay effect. We then compare these data to the predictions of TS99 and discuss the results. In Section 5, we briefly summarize our work.

Regular pulsar timing observations of PSRA in NGC 6752 have been carried out since 2000 September with the Parkes 64 m radio telescope at a central frequency of 1390 MHz, using the central beam of the 20 cm multibeam receiver (Staveley-Smith et al. 1996) or the H-OH receiver. The hardware system is the same as that used in the discovery observations (Paper I). The effects of interstellar dispersion are minimized by using a filterbank having 512×0.5 MHz frequency channels for each polarization. After detection, the signals from individual channels are added in polarization pairs, integrated, 1-bit digitized every 80 μs (125 μs in earlier observations), and recorded to magnetic tape for offline analysis. We synchronously folded the data at the pulsar period using the program DSPSR (van Straten & Bailes 2011) with a subintegration time of 1 minute and dedispersion in groups of 64 channels to give 8 frequency sub-bands. We visually inspected each file to extract the maximum number of pulse times of arrivals (ToAs), depending on integration time and interstellar scintillation which causes large fluctuations in the detected flux density. We considered a ToA reliable if the pulse profile had a signal-to-noise ratio greater than 5 and the pulse was visible. This approach gave us ∼1000 ToAs at different frequencies within the band of the receivers and consequently allowed us to fit for the dispersion measure and its first derivative. We calculated the ToAs by fitting a template profile to the observed mean pulse profiles and we analyzed them using the program TEMPO2 (Hobbs et al. 2006) with the DE405 solar system ephemeris and the TT(TAI) reference timescale. Table 1 summarizes the best-fit values and 1σ uncertainties from TEMPO2 (everywhere in the paper all reported measures are quoted with their 1σ uncertainty, and all reported ranges are at the same confidence level). The timing residuals for the fit which have a χ² of 989.3 (with 989 degrees of freedom) are shown in Figure 1, against MJD in the upper panel and against orbital phase in the lower panel. We determined the values and the uncertainties for the Shapiro delay parameters through a Bayesian analysis (see Section 3).

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Table 1. Measured and Derived Parameters for PSR J1910−5959A

Parameter	PSR J1910−5959A
R.A. (J2000)	19:11:42.75562(3)
Decl. (J2000)	−59:58:26.9029(3)
μαcos δ (mas yr−1)	−3.08(6)
μδ (mas yr−1)	−3.97(6)
P (ms)	3.26618657079054(9)
$\dot{P}$ (10−21)	2.94703(14)
DM (pc cm−3)	33.6998(16)
DM1 (pc cm−3 s−1)	0.00136(36)
Binary model	DD
Porb (days)	0.83711347691(3)
aPsin i (lt-s)	1.2060418(7)
T0 (MJD)	51919.206480057(55)
M(a)c (M☉)	0.180(18)
sin i(a)	⩾0.9994
i(a) (deg)	⩾88°
f(MPSR) (M☉)	0.00268782589(5)
μ (mas yr−1)	5.02(6)
Position angle (deg)	217.9(7)
Ref. Epoch (MJD)	51920.000
MJD range	51710–55911
Number of TOAs	1003
Rms residuals (μs)	5.3

Notes. Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. Numbers enclosed in parenthesis represent the 1σ uncertainties on the last significant digit for all parameters, either measured and derived, presented in this table. The values and relative uncertainties for the parameters that describe the Shapiro delay (indicated with the apex a) have been obtained through a Bayesian analysis of our timing results. The proper motion's position angle is measured counterclockwise starting from the north direction.

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The new positional and rotational parameters at the reference epoch are all consistent with those reported in Paper I and Paper II at a confidence level of 2σ, with the only exception being the proper motion, whose components in right ascension and declination are consistent at the 4σ and 6σ levels, respectively. The value for the first derivative of the dispersion measure (DM1) is of the same order of magnitude of similar determinations for other MSPs. The consistency of the orbital parameters will be discussed in Section 3.

3.1. Method

The orbital solutions presented and discussed in Papers I and II were based on the ELL1 binary model, which is suitable for systems with small orbital eccentricities (N. Wex 1998 unpublished work; see Lange et al. 2001). In Paper II, an orbital eccentricity e = (3.4 ± 0.6) × 10⁻⁶ was reported at a longitude of periastron ω = 90° ± 10°. The values obtained for e and in particular for ω were already commented as a possible misinterpretation of a Shapiro delay (Paper II). By applying the same binary model of Paper II to the longer data span presented in this work, we obtain e = (3.5 ± 0.5) × 10⁻⁶ at a longitude of periastron ω = 95° ± 7°. Since the uncertainties on these two parameters did not decrease as expected for the case of a pure Keplerian ELL1 model, we carefully reinvestigated the possible presence of a Shapiro Delay.

We used the DD binary model (Damour & Deruelle 1985, 1986) and included the two parameters that describe the Shapiro delay, namely, the range r and the shape s. These two parameters are directly related to the companion mass M_C and the orbital inclination i through the simple relations r = T_☉M_C (T_☉ = GM_☉/c³, where G is Newton's universal gravitation constant and c is the speed of light) and s = sin i (see, e.g., Lorimer & Kramer 2004). Figure 2 displays our attempts to detect the first and third harmonics of the Shapiro delay in our timing residuals. We recall (e.g., Lorimer & Kramer 2004) that the first harmonic is obtained by fitting for all parameters, including the ones describing the Shapiro delay, then resetting M_C = 0.0 and sin i = 1.0 while keeping all other parameters to their best-fit values, and finally by plotting the resulting timing residuals. The third harmonic is obtained by fitting for all parameters but the Shapiro delay ones, which remain set to the values M_C = 0.0 and sin i = 1.0, then plotting the residuals for this best-fit solution obtained in this way. While the first harmonic is still consistent with the response for an elliptic orbit, the third harmonic is a unique signature of the Shapiro delay. In the upper panels of Figure 2, timing residuals show some evidence of the harmonic structure, while in lower panels, where residuals have been rebinned in 20 orbital phase bins, the two harmonics are clearly recognizable. As a further comparison, in the lower panels we plot (solid lines) the theoretical curves for the Shapiro delay with which our rebinned residuals are in satisfactory agreement.

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As an additional sanity check, we have checked the consistency of the three Keplerian parameters that are common to the two different orbital models presented in Paper II and in this work. The values for P_B are fully consistent to each other, and the same is true comparing T_ASC in Paper II and T₀ here. We recall that in the case of zero eccentricity orbits, the periastron is defined as coincident to the ascending node. The only parameter for which we now find an inconsistency at a level of about 10σ with respect to Paper II is the orbit projected semimajor axis a_Psin i. However, because of the change in the adopted binary model (from ELL1 to DD), this parameter cannot be directly compared. The quantity to be compared between is a_Psin i/(1 + e). From Paper II, we obtain a_Psin i/(1 + e) = 1.2060425 ± 0.0000007, while in this work it results a_Psin i = 1.2060420 ± 0.0000008, as the orbital eccentricity has been now set to zero. The two values are then in full agreement.

We obtained the values for the companion mass and the inclination angle through a Bayesian analysis on our timing residuals. The method is described in Splaver et al. (2002). We chose the a priori probability density functions as follows.

• 1.  
  A flat distribution for cos i in the range 0 ⩽ cos i ⩽ 1, as we assumed a randomly oriented orbit in space.
• 2.  
  A flat distribution for M_C in the range 0.0 M_☉ ⩽ M_C ⩽ 0.5 M_☉, chosen for consistency with the values of this parameter resulting from the study of the radial velocity curve of the companion (Bassa et al. 2006; Cocozza et al. 2006).

Figure 3 displays in panel (a) the Δχ² map obtained after calculating the χ² values in a 2048 × 2048 grid for the parameters of our interest. In panels (b) and (c), the posterior probability density functions for cos i and M_C, respectively, are reported. From this analysis, we derived a lower limit for the orbital inclination sin i ⩾ 0.9994, i.e., i ⩾ 88° and the value for the companion mass M_C = 0.180 ± 0.018 M_☉.

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3.2. Implications

TS99 presented detailed numerical calculations on the non-conservative evolution of close binary systems with a low-mass (M_D < 2.0 M_☉) donor star, a 1.3 M_☉ accreting neutron star, and an orbital period P_B ⩾ 2 days at the beginning of the LMXB phase. The evolution of LMXBs has already been modeled by Pylyser & Savonije (1988, 1989), who predict that binaries with starting periods P_B ⩾ P_bif ≃ 2 days increase their orbital separation, hence their orbital period, during the mass transfer (diverging systems), while the ones with P_B ⩽ P_bif decrease their orbital separation (converging systems). In this second case, a simple argument shows that the binary period decreases if $|\dot{M}_{\rm tot}|/M_{\rm tot}\,<\,|\dot{a}|/a$ and increases if $|\dot{M}_{\rm tot}|/M_{\rm tot}\,>\,|\dot{a}|/a$, where M_tot is the total mass of the binary and a is its orbital separation.

TS99 focused their attention on diverging LMXBs and they obtain a positive correlation between the orbital period and the final mass of the HeWD companion, i.e., its mass increases with the observed orbital period, and a negative correlation between the orbital period and the mass of the pulsar. In order to compare the theoretical results of TS99 to the observed systems, we searched the literature for low-mass binary pulsars for which a reliable measurement of both the pulsar and the companion mass has been obtained. These values are reported in Table 2 jointly with the binary period and the method used to obtain the masses.

The masses of the two orbiting objects have been determined using Shapiro delay for 6 binaries out of the 12 reported in Table 2 (indicated with the label "r,s"). As we already discussed in Section 3.2, this kind of measurement does not need any a priori hypothesis, hence these six objects can be used to perform six firm and independent tests on the proposed models. The masses for the remaining objects have been determined as follows.

• 1.  
  PSR J1012+5307 (van Kerkwijk et al. 2005) PSR 1738+0333 (Antoniadis et al. 2012) and PSR B1957+20 (van Kerkwijk et al. 2011). The binary companion has been observed in phase-resolved spectroscopy observations in the optical band. This observing mode leads to the determination of the companion mass function. Pulsar timing gives the pulsar mass function and the knowledge of these two quantities leads to the determination of the mass ratio of the system. The companion mass has been determined by comparing the observed spectroscopic properties to theoretical models for the atmosphere of HeWDs.
• 2.  
  PSRs J1853+1303, J1910+1256, and J2016+1948 (Gonzalez et al. 2011). The timing analysis gives a measurement of the proper motion and of the first derivative of the pulsar projected semimajor axis (d(a_Psin i)/dt). This second quantity changes because of the changing orbital inclination due to proper motion (Kopeikin 1996; Arzoumanian et al. 1996). Since this also depends on the unknown angle between the ascending node and proper motion directions, the most probable orbital inclination has been determined using Monte Carlo simulations. The companion mass is assumed to be that predicted by the TS99 calculation, while the pulsar mass has been obtained by combining the companion mass with the statistically determined orbital inclination and the pulsar mass function.

In Figure 4, we plot the logarithm of the binary period versus companion mass (upper panels) and the pulsar mass (lower panels). We plotted all objects reported in Table 2 in the left panels, while in the right ones we only show those binaries for which the masses have been determined from measured Shapiro delays. We marked with a triangle the position of PSRA in these plots to distinguish it from the field systems which are denoted by diamonds.

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In the left upper panel of Figure 4, the solid, dashed, and dotted lines represent the predictions of the M_C–P_B correlation by TS99 (Equations (20) and (21)). With the exception of the squared point in the left bottom part of the plot, all other points well agree with the theoretical lines by TS99. Intriguingly, the extrapolations of the results of TS99 to binary periods lower than 2 days are also in agreement with the observations of binaries with P_B ⩽ 2 days, i.e., systems whose progenitors were converging LMXBs. This means that the hypothesis assumed by TS99 to model the diverging LMXBs can predict rather well the final HeWD mass also for converging systems. The three points highlighted by a diamond shape represent the binaries for which the companion mass was obtained from one of the TS99 models (Gonzalez et al. 2011), so of course they fall on that line in the M_C–P_B plot. The squared point represents the position of PSR B1957+20, a black widow system. For this latter system, we cannot compare the current companion mass with the predictions by TS99 because of the mass loss the companion underwent after the formation of the MSP+WD system. Nevertheless, for the binary period of this system, TS99 predicts a companion mass ∼0.18 M_☉, i.e., higher than the observed one. This is consistent with the hypothesis that this system obeyed the TS99 M_C–P_B correlation at the epoch of the formation of the MSP+WD binary.

We note that the overall agreement of the predicted M_C–P_B correlation with the available measurements has already been discussed by various authors (e.g., van Kerkwijk et al. 2005; Lorimer 2008). However, these comparisons were based on a variety of (often) indirect methods for determining M_C, in many cases with rather large confidence intervals for the value for M_C. In the upper right panel of Figure 4, we display a test for the M_C–P_B correlation entirely based on measures obtained via the detection of the Shapiro delay. The theoretical models for the M_C–P_B correlation by TS99 result in agreement to all six test binaries of our sample, including PSRA, the only binary likely associated with a globular cluster.

In the lower left plot of Figure 4, 8 of 11¹¹ objects show a linear correlation between M_P and P_B, while the remaining 4, highlighted with a circle, form a separate group. A comparison between this panel and Figure 4(b) in TS99 indicates that the theoretical predictions overestimate by about 0.5 M_☉ the pulsar mass for a given orbital period in the entire range considered by TS99. If a real correlation exists between M_P and P_B for this class of binaries, it is not the one predicted by TS99 as these authors already pointed out in their comparison with the observed systems.¹²

The solid line is a fit for all plotted points but the circled ones modeling the M_P–P_B correlation by using the following function:

$$\begin{equation} \frac{M_P}{M_\odot }\,=\,A+B\log _{10}\left(\frac{P_B}{{\rm 1\,day}}\right) \end{equation} \tag{ 1 }$$

and we obtain A = 2.180 ± 0.005 and B = − 0.475 ± 0.002 (χ² = 0.58 with 5 degrees of freedom).

In the bottom right panel of Figure 4, the six binaries which represent our firm tests for TS99 are equally divided into two groups: three of them obey the M_P–P_B correlation mentioned above, while the other three do not. The binaries in this second group also have binary periods lower than the bifurcation period P_bif ≃ 2 days theoretically found by Pylyser & Savonije (1988, 1989), hence their progenitors were converging LMXBs, while the other three have binary periods higher than P_bif, so it is more likely that their progenitors were diverging LMXBs.

The quantitative discrepancy between the TS99 predictions and the observed systems in the M_P–P_B diagram, combined with the agreement in the M_C–P_B diagram, indicates that in TS99 the amount of mass released by the companion is well predicted, but the amount of matter captured by the neutron star is overestimated. This means that the discrepancies between the theoretical and observed M_P–P_B diagrams result from an incorrect modeling by TS99 of the mechanisms (one or more) that prevent the neutron star from capturing all the matter released by its companion and are responsible for the mass loss that the binary system undergoes in its LMXB phase. Two mechanisms have been proposed, namely, the radio ejection model (Burderi et al. 2001) and the propeller model (Tauris 2012). Since our test binaries divide themselves in two separate groups, which in turn should have experienced different evolutionary paths for their LMXB progenitors, it is tempting to suggest that the mechanism responsible for the unaccreted matter is different in the two cases of converging and diverging LMXBs, or that in one case only one mechanism takes place at some point of the evolution, while in the other case both mechanisms play a non-negligible role.

The position in the M_P–P_B plot (bottom left) of PSR B1957+20 is apparently at odds with this picture. Its orbital period indicates that its progenitor was a converging LMXB, but it agrees with the possible M_P–P_B correlation for binaries whose progenitors were diverging LMXBs. A first possibility is that the pulsar mass in this system is overestimated. In fact, as reported above, M_P was obtained using indirect methods relying on modeling of the outer layers of the companion star. If the values for M_P of PSR B1957+20 is confirmed by future observations, we could speculate that the family of converging LMXBs must be in turn subdivided in two groups. For the first group, the mechanism(s) preventing the accretion of mass are similar to the diverging systems and the resulting HeWD–MSP binaries obey the same M_P–P_B correlation. The second group is instead characterized by peculiar conditions that make the loss of matter from the binary more efficient. Detailed numerical simulations are required to clarify this point.

In this work, we have presented a timing solution for the binary MSP PSR J1910−5959A based on Parkes timing data with a data span of 12 years. Our new solution gives improved values for the astrometric and rotational parameters compared to those reported in Papers I and II, and contains a new description of the orbital motion including the first detection of the relativistic Shapiro delay for a low-mass MSP in a globular cluster. The measurement of this effect allowed us to determine reliable masses for the two orbiting stars. We have compared the values for the mass and the radius of the companion to the theoretical mass–radius relationships obtained for different chemical compositions, but the current uncertainties on these parameters prevent us from firmly establishing whether PSR J1910−5959A belongs to NGC 6752 or not. Finally, we compared the numerical results on the evolution of LMXB systems obtained by TS99, which predict the presence of a correlation between M_C and P_B and between M_P and P_B for binary systems where an MSP orbits a low-mass HeWD companion, to the sample of these objects for which the masses of the two stars have been measured using Shapiro delay. Observations confirm the M_C–P_B correlation obtained by TS99 and show a possible M_P–P_B correlation for a subset of the sample which is different to the one obtained by TS99. The numerical calculations by these authors overestimate the pulsar mass by about 0.5 M_☉. These results confirm that the amount of mass lost by the binary during the evolution in LMXB is always larger than predicted by TS99. Moreover, they suggest that the low-mass binary pulsars with a WD companion can be split in two groups, characterized by significantly different efficiencies in the mechanisms responsible for the loss of matter during their evolution in the LMXB phase.

A.C., A.P., and N.D. acknowledge support for this research provided by INAF, under the National Research Grant PRIN-INAF 2010, awarded with DP 28/2011. The Parkes radio telescope is part of the Australia Telescope, which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. We thank the colleagues who assisted with the observations discussed in this paper.