HIGH-PRECISION TIMING OF FIVE MILLISECOND PULSARS: SPACE VELOCITIES, BINARY EVOLUTION, AND EQUIVALENCE PRINCIPLES

All pulsars were observed with the 305 m Arecibo telescope in Puerto Rico. The Wideband Arecibo Pulsar Processors (WAPPs; Dowd et al. 2000) were used to observe all the pulsars. Three of the four WAPPs were used for most pulsars, except PSR J2016+1948 for which all four WAPPs were used in some observations. They were operated in online folding mode with 32 μs sampling and 192 lags near 1400 MHz and 96 lags at 2700 MHz (for PSR J2016+1948 a sampling time of 128 μs and 128 lags were used). The Arecibo Signal Processor (ASP; Demorest 2007) was used for all pulsars except PSR J2016+1948. ASP provides 0.25 μs complex sampling in two orthogonal polarizations. These data were coherently de-dispersed in software using 16 or 24 frequency channels, each with 4 MHz bandwidth. The polarizations were later summed and the signal was folded at the pulsar period. The ASP observations were flux calibrated with a pulsed noise diode of known strength and, when available, with observations of a standard flux calibration source. A typical observing session involved collecting multiple integrations of 1 or 3 minutes long on each pulsar with ASP and the WAPPs for a total of up to 30 minutes of data. The data from the short integrations were aligned and summed using a preliminary timing model. For the WAPPs, all data were summed to obtain at single profile for each observation. For ASP, a separate profile was obtained at each frequency channel for each observation.

For PSR J2016+1948, we also used data from the Penn State Pulsar Machine (PSPM; Cadwell 1997), an analogue filterbank with 128 × 60 kHz frequency channels. The power level for each 128 channels was sampled every 80 μs and stored to tape. The data were subsequently folded and aligned multiple times as the ephemeris for the pulsar was being refined. The PSPM and WAPP data for this pulsar were summed every few minutes and multiple profiles were obtained for most epochs.

To provide a long-time baseline and improve the measured proper motion for PSR B1953+29, we also used data taken with the Mark II system (Rawley 1986), a dual-polarization 32 × 30 kHz filterbank spectrometer. Here, the outputs from opposite polarizations were summed and sampled at the pulsar period. Data were averaged over intervals of one to two minutes and stored for off-line processing. A total of around one hour of data was collected at each epoch and the data summed to obtain one profile for each observation.⁹

We used the 64 m Parkes telescope in NSW, Australia, to observe pulsars PSR J1853+1303, PSR J1905+0400, and PSR J1910+1256. These observations were taken using the Parkes analog filter bank centered at 1390 MHz with 512 × 0.5 MHz frequency channels sampled every 0.25 ms. Two polarizations were recorded and summed in hardware for each frequency channel. The data were subsequently folded offline using a preliminary ephemeris and summed to obtain a single profile for each observation.

A time of arrival (TOA) was found for each observation by cross-correlating the profiles with a high signal-to-noise standard template (Taylor 1992). The recorded observatory times were corrected to UTC time by using data from GPS satellites. The JPL DE405 ephemeris (Standish 2004) was used for barycentric corrections. The software package TEMPO¹⁰ was used to find the timing solution for each pulsar by including astrometric, binary, and spin parameters as needed to arrive at a phase-connected solution (where every rotation of the star over the span of the observations is accounted for). In order to fit for any instrumental or standard template profile differences, we fit for arbitrary time offsets between each instrument (for ASP we have also allowed for time offsets between each 4 MHz channel to fit for any profile changes across its bandwidth). A change in dispersion measure (DM) over time was detected for PSR J2016+1948 and marginally for PSR J1905+0400 and PSR B1953+29 (see Table 2). Finally, the measured uncertainties were scaled by a small telescope-dependent amount to ensure a timing fit with χ²_ν ≃ 1.

Table 2. Measured and Derived Parameters for the Observed Pulsars

Parameter	PSR J1853+1303	PSR J1905+0400	PSR J1910+1256	PSR B1953+29	PSR J2016+1948
Right ascension (R.A.), α (J2000.0)	18:53:57.319174(8)	19:05:28.273436(16)	19:10:09.701479(8)	19:55:27.87600(3)	20:16:57.44349(6)
Declination (decl.), δ (J2000.0)	13:03:44.0784(2)	04:00:10.8830(6)	12:56:25.5074(3)	29:08:43.4659(5)	19:47:51.5882(12)
Epoch (MJD)	54000	53700	54000	54500	53000.0
Data span (MJD)	52606.1–55134.8	51492.2–54808.8	52602.2–55171.8	46112.6–55134.9	52456.2–55392.2
Proper motion in R.A., μα = $\dot{\alpha } \cos \delta$ (mas yr−1)	−1.68(7)	−3.80(18)	0.21(10)	−0.9(1)	1.28(26)
Proper motion in decl., $\mu _\delta = \dot{\delta }$ (mas yr−1)	−2.94(12)	−7.3(4)	−7.25(12)	−4.1(1)	2.83(34)
Annual parallax, π (mas)	1.0(6)	<2.5a	<0.7a	<7a	<4.5a
Spin frequency, ν (s−1)	244.3913778653740(15)	264.242346143483(16)	200.658805375034(1)	163.04791306911(2)	15.3987376281305(13)
Spin frequency derivative, $\dot{\nu }$ (s−2)	−5.2060(5) × 10−16	−3.425(1) × 10−16	−3.900(2) × 10−16	−7.901(3) × 10−16	−9.4997(14) × 10−17
Dispersion measure, DM (pc cm−3)	30.5701(6)	25.6923(12)	38.0701(8)	104.501(3)	33.8148(16)
First DM derivative, $\dot{\rm DM}$ (pc cm−3 yr−1)	<|4 ×10−4|a	−1.1(7) × 10−3	<|6 ×10−4|a	−4.7(2.5) × 10−3b	−1.35(12) × 10−3b
Orbital period, Pb (days)	115.65378643(2)		58.466742029(12)	117.34909728(4)	635.02377864(7)
Project semimajor axis, x (lt-s)	40.7695198(3)		21.1291036(3)	31.4126903(8)	150.773037(2)
Rate of change of x, $\dot{x}$ (ls-s s−1)	1.7(7) × 10−14		−1.8(5) × 10−14	<4 × 10−14a	8.3(14) × 10−14
Eccentricity, e	2.3691(12) × 10−5		2.3018(3) × 10−4	3.3025(5) × 10−4	1.47981(2) × 10−3
Longitude of periastron, ω (deg)	346.60(4)		106.014(9)	29.485(8)	95.6398(7)
Epoch of periastron, T0 (MJD)	54046.78(1)		54079.318(1)	54444.267(2)	52818.648(1)
Number of TOAs	695	549	633	316	2210
Weighted rms residual (μs)	1.5	5.4	1.8	3.8	11.5
Derived Parameters
Spin period, P (ms)	4.09179744490025(2)	3.78440484691321(7)	4.9835840178364(1)	6.1331666053350(1)	64.940389248427(5)
Spin period derivative, $\dot{P}$ (s s−1)	8.7163(8) × 10−21	4.905(2) × 10−21	9.687(4) × 10−21	2.9734(1) × 10−20	4.0063(6) × 10−19
Surface magnetic field, B (G)	1.9 × 108	1.4 × 108	2.2 × 108	4.3 × 108	5.2 × 109
Spin-down luminosity, $\dot{E}$ (1033 erg s−1)	5.1	3.5	3.1	5.1	0.058
DM-derived distance, dDM (kpc)	2.1	1.7	2.3	4.64	2.5
Characteristic age, τc (Gyr)	7.4	12.2	8.1	3.3	2.6
Mass function, f1 (M☉)	0.00543963576(7)		0.002962840(12)	0.00241678837(2)	0.00912586(4)
Minimum companion massc, m2 (M☉)	0.24		0.19	0.18	0.29
Total proper motion, μ (mas yr−1)	3.39(11)	8.24(36)	6.98(14)	4.2(1)	3.11(32)
Galactic angle of proper motiond, Θμ	274°	270°	210°	197°	277°

Notes. Values in parentheses are uncertainties in the last digits shown, which are twice the formal errors quoted by TEMPO after scaling the TOA uncertainties to obtain χ²_ν ≃1. Right ascension values are in hours, minutes, and seconds, and declination values are in degrees, arcminutes, and arcseconds. For all pulsars, the DE405 ephemeris was used and the recorded observatory times were corrected to TT(BIPM). ^aValue shown represents a Δχ² ∼ 6.6 from best fit, representing a ∼3σ limit (Avni 1976). ^bFor PSR J2016+1948, a second DM derivative (${\rm {\ddot{D}M}}$) was also measured with a value of 3.8(1.4) × 10⁻⁴ pc cm⁻³ yr⁻². A less significant value for ${\rm {\ddot{D}M}}$ was also measured for PSR B1953+29 giving −2.1(1.6) × 10⁻⁴ pc cm⁻³ yr⁻². ^cAssuming a pulsar mass of m₁ = 1.35 M_☉. ^dClockwise from Galactic north.

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The high precision obtained by our timing study allowed us to measure a statistically significant value for the proper motions of all five pulsars (see Table 2). We have used these measurements to study the velocity distribution of MSPs, both isolated and those in binary systems. The pulsar population in general has been found to have large space velocities with a mean of ∼300 km s⁻¹ (Lyne & Lorimer 1994; Hobbs et al. 2005). Recycled MSPs appear to be on the low end of the velocity distribution, with a mean of ∼90 km s⁻¹. In addition, no significant difference has been found between the velocity distributions of isolated MSPs and those still in binary systems (Hobbs et al. 2005; Lommen et al. 2006) despite the fact that an additional evolutionary stage (the disruption of the binary) has occurred in the former.

Now we revisit the velocity distribution of MSPs, which we take to be those with periods of P < 0.01 s and are therefore fully recycled. From the new timing solutions presented in Table 2, only PSR J2016+1948 is not a fully recycled MSP and its implied two-dimensional (2D) velocity of 96 km s⁻¹ (at the implied DM distance of 2.5 kpc) was not included in the following analysis. PSR J1905+0400 studied here is particularly important, as it is one of only 10 isolated MSPs with measured proper motions. Our sample then consists of 10 isolated MSPs and 27 binary MSPs for a total of 37 pulsars. We have combined the measured proper motions with the available distance estimates to calculate the pulsars' 2D space velocities in their respective local standard of rest at the pulsar location. To do this we have corrected for solar motion and used a peculiar velocity for the Sun of V_☉ = 13.4 km s⁻¹ (Dehnen & Binney 1998). We have also assumed a flat Galactic rotation curve with a galactocentric distance for the Sun of R_☉ = 8 kpc and a Galactic rotation velocity of 222 km s⁻¹ (Eisenhauer et al. 2003; Dehnen & Binney 1998). A flat Galactic rotation curve is thought to be a good approximation for distances from the Galactic center of >3 kpc (Olling & Merrifield 1988; the pulsars we used have distances to the Galactic center of >4 kpc). The resulting 2D velocities in the pulsars' standard of rest after correcting for solar and Galactic motion, V_2D, are shown in Table 3.

Most pulsars have distance estimates from timing measurements of their DM combined with a model of the free electron distribution in the Galaxy (Cordes & Lazio 2002). In general, distances derived using this method are thought to have a ∼25% error, implying a minimum similar error on the estimated velocities. For individual pulsars, the distance error could be much larger than that. The errors on the estimated pulsar velocities shown in Table 3 were derived using Monte Carlo simulations with 10,000 runs per pulsar. For these simulations, pulsar distances were drawn from Gaussian distributions centered on the values listed in Table 3 with a width of 25% of the central value¹¹ (for pulsars where non-DM distances are available the corresponding distance errors were used). Pulsar proper motions were then drawn using Gaussian distributions with central values and widths derived using the values listed in Table 3. In practice, the largest error contribution to the estimated pulsar velocities are the associated distance errors. We then used the velocities in Table 3 to study the velocity distribution of MSPs.

Figure 4 shows the normalized histograms of the 2D velocities of binary (solid line) and isolated MSPs (bold dotted line) in our sample. The average velocities are found to be 108 ± 15, 113 ± 17, and 93 ± 20 km s⁻¹ for all MSPs, binary MSPs, and isolated MSPs, respectively.¹² The average 2D velocities without correcting to the pulsars' standard of rest are 88 ± 12, 96 ± 15, and 68 ± 16 km s⁻¹ for all, binary, and isolated MSPs, respectively. The updated velocities are consistent with previous work: Hobbs et al. (2005) reported 2D uncorrected velocities for binary and isolated MSPs of 89 ± 15 and 76 ± 16 km s⁻¹, respectively, with Lommen et al. (2006) and McLaughlin et al. (2005) reporting similar values. We then find that the average velocities of binary and isolated MSPs are consistent with each other. To test whether the two samples are consistent with arising from the same distribution we use the Kolmogorov–Smirnov (K-S) test (Massey 1951).¹³ A K-S test of the two corrected velocities results in a probability of 62% that they are drawn from the same distribution. For the uncorrected 2D velocity measurements, the two distributions have a K-S probability of 75% that they are drawn from the same distribution.

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We therefore conclude that there is no statistically significant difference between the velocity distributions of isolated and binary MSPs with the current statistics. However, we also note that due to selection effects our sample is biased toward nearby, low-velocity pulsars. It is therefore possible that the lack of difference between the velocity distribution of isolated and binary MSPs is due to our observing the low-velocity tail of these distributions, which in reality could be quite different. Higher number statistics (particularly for isolated MSPs) will allow for a more detailed study of such effects in the future. In addition, more distant MSPs are now being discovered in current surveys (e.g., in addition to PSR J1903+0327 in Table 3, the velocities for two distant MSPs will also be published by J. S. Deneva et al. 2011, in preparation). While measuring the proper motions of distant objects will most likely require large amounts of telescope time, they represent significant additions to our sample.

Furthermore, we note that Tauris & Bailes (1996) presented the expected space velocities of binary MSPs using various evolutionary models. In their simulations, binaries with shorter periods tend to have larger velocities since the final velocity of the system depends on the separation of the components at the time of the SN explosion. However, this correlation is fairly weak and asymmetries in the explosion would easily wipe out this effect. Hobbs et al. (2005) found no evidence for a correlation between the velocity of binary MSPs and their binary periods.¹⁴ We now briefly revisit this idea and in Figure 5 we plot the binary periods versus implied velocity for the binary pulsars listed in Table 3. This figure should be compared to the plots in Figure 2 of Tauris & Bailes (1996). For a system with P_b < 2 days and P_b > 2 days, we find average 2D velocities of 135 ± 52 km s⁻¹ and 107 ± 14 km s⁻¹, respectively (uncorrected 2D velocities have averages of 120 ± 45 km s⁻¹ and 85 ± 11 km s⁻¹).

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We therefore find no significant difference in the velocities of short- and large-period binary MSPs. However, we caution that only a handful of the former systems are known. While the very large velocity of PSR B1957+20 could be explained by these models using asymmetries in the SN explosion, they would have a particularly hard time explaining the small implied velocities for PSR J0751+1807 and PSR 2051−0827 given their very short orbital periods. In addition, we did simulations using the Tauris & Bailes (1996) binary period–velocity relationship, taking into account the effect of random projections toward us of these velocities. We find that the large scatter in the relationship and the random projections of these velocities would most likely wash out any effect. Therefore, while we find no evidence that the relationship is present (and this might indeed be very difficult to achieve, even if it exists), at the same time we cannot rule it out. It is clear that additional work is needed to understand the evolution of binary MSPs. Obtaining additional velocity measurements for these pulsars will help to constrain evolutionary models.

For PSR J1910+1256 and PSR J2016+1948, we were able to measure a change in the projected semimajor axis, $\dot{x}$ = dx/dt (see Table 2). For PSR J1853+1303, the measurement of $\dot{x}$ was marginal and will be discussed at the end of this section. For PSR B1953+29 only an upper limit was measured. Here we define x ≡ a₁sin i/c, where a₁ is the semimajor axis of the pulsar orbit, i is the inclination angle of the angular momentum vector of the orbit relative to the Earth-pulsar line of sight (LOS), and c is the speed of light. The measured values for $\dot{x}$ are −1.8(5) × 10⁻¹⁴ and 8.3(14) × 10⁻¹⁴ for PSR J1910+1256 and PSR J2016+1948, respectively. In principle, a non-zero value for $\dot{x}$ could arise from a change in a₁, i, or a combination of the two. However, we argue that the measured values likely arise due to the pulsars' high proper motion inducing a change in our LOS to these binaries.

In the case that a change in a₁ is being observed, GR predicts a value of $|\dot{a_1}|$ ∼ 5 × 10⁻²³ and ∼10⁻²⁴ for PSR J1910+1256 and PSR J2016+1948, respectively (see Peters 1964 for the required expression). These values are many orders of magnitude below the observed value of $\dot{x}$. In addition, for typical binary astrophysical processes, a non-zero value for $|\dot{a_1}/a|$ is expected to have a similar order of magnitude as $|\dot{P_b}/P_b|$. No significant value for $\dot{P_b}$ was found for any of our pulsars, but allowing for a measurement in our timing solution results in a value of $\dot{P_b}$ = −2(4) × 10⁻¹¹ and −1(2) × 10⁻⁹ s s⁻¹ for PSR J1910+1256 and PSR J2016+1948, respectively. Using the values for P_b listed in Table 2, we find limits of $|\dot{P_b}/P_b|<$ 1.2 × 10⁻¹⁷ s⁻¹ and <5 × 10⁻¹⁷ s⁻¹ for PSR J1910+1256 and PSR J2016+1948, respectively. Again, these values are a few orders of magnitude lower than expected from the measured values of $\dot{x}$.

We therefore propose that the observed values of $\dot{x}$ must arise from apparent changes in the orbital parameters due to the proper motion of the binaries (Arzoumanian et al. 1996; Kopeikin 1996). The strength of this geometric effect can be calculated using

$$\begin{equation} \dot{x}=1.54\times 10^{-16}x\mu \cot {i}\sin {\theta } \;\; {\rm s\, s^{-1}}, \end{equation} \tag{ 1 }$$

where x is the projected semimajor axis in units of seconds, μ is the total proper motion of the system in units of mas yr⁻¹, and θ is the unknown angle between the position angle of the proper motion and the position angle of the ascending node of the pulsar's orbit. The measured values of $\dot{x}$ can then be used to constrain the inclination angle of the system i. Following Nice et al. (2001), we have used a Monte Carlo simulation to constrain the values of i that satisfy Equation (1) for both PSR J1910+1256 and PSR J2016+1948, resulting in 1σ values of 44°(36°–52°) and 36°(27°–45°), respectively.

In addition, numerical studies of the evolution of neutron star binaries in long-period orbits with low-mass WD companions point to a relationship between the final orbital period, P_b, and the mass of the companion, m₂ (Rappaport et al. 1995; Tauris & Savonije 1999). An overall agreement with these results has been found in the available data, although these relationships appear to overestimate m₂ for systems with long periods and provide conflicting results for systems with short periods (Stairs et al. 2005).

Keeping these caveats in mind, we used the P_b–m₂ relationship found by Tauris & Savonije (1999), together with the inclination angle constrains from $\dot{x}$, to provide constraints on the masses of the PSR J1910+1256 and PSR J2016+1948 systems. The derived constraints on the companion masses are m₂ = 0.30–0.34 M_☉ and 0.43–0.47 M_☉ for PSR J1910+1256 and PSR J2016+1948, respectively. These values and those implied by the mass functions are shown in Figure 6. Restricting m₂ to lie in the values implied by the P_b–m₂ relationship and using the inclination angles from $\dot{x}$ we find 1σ values for the pulsar masses of m₁ = 1.6 ± 0.6 M_☉ and 1.0 ± 0.5 M_☉ for PSR J1910+1256 and PSR J2016+1948, respectively.

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For PSR J1853+1303, the measured value of $\dot{x}$ is marginal (see Table 2) but can still be used to derive initial constraints on the system masses. Following the same procedure as outlined above results in a 1σ confidence interval for the inclination angle of this system of 48°(33°–58°). In addition, the P_b–m₂ relationship produces companion masses of m₂ = 0.33–0.37 M_☉. Combining these results, we derive 1σ values for the mass of PSR J1853+1303 of m₁ = 1.4 ± 0.7 M_☉.

The derived m₁ values are not very constraining, though fully within the expected mass ranges for neutron stars. Given that PSR J2016+1948 is only one of three WBMSPs with P_b > 200 days (see Table 4), further constraining the masses of this system by independent measurements and continued timing can provide a valuable constrain to binary pulsar evolution models.

We have used the improved timing parameters for the four WBMSPs studied here, in addition to recently discovered systems, to update important tests of GR and other theories of gravity. In particular, we have modeled the forced eccentricity that would be imparted on the binary systems due to violations of the SEP using the parameter Δ, and the forced eccentricity that would be imparted due to violations of Lorentz invariance/momentum conservation using the parameter $\hat{\alpha }_3$. In GR, both parameters are predicted to be identically zero and $\hat{\alpha }_3$ is also predicted to be zero in most other theories of gravity.

For the SEP test parameter Δ, the additional, forced eccentricity imparted on the binary orbit is expected to be of the form (Damour & Schäfer 1991)

$$\begin{equation} |{\bf e}_{F,\Delta }| = \Delta \frac{|{\bf g}_\perp | c^2}{2FGM(2\pi /P_b)^2}, \end{equation} \tag{ 2 }$$

where c is the speed of light, F is unity in GR and a function of m₁ and m₂ in alternate theories, G is Newton's constant in GR, M = m₁+m₂, P_b is the binary period, and |g_⊥| is the projection of the Galactic acceleration vector onto the orbital plane at the location of the pulsar. Here, the total observed eccentricity is then predicted to be e_obs = e_N + e_{F, Δ}, with the "natural" eccentricity e_N and the angle θ between e_N and e_{F, Δ} being additional, unknown parameters.

For the Lorentz invariance/momentum conservation test parameter $\hat{\alpha }_3$, the forced eccentricity added to the binary orbit is expected to be given by (Bell & Damour 1996)

$$\begin{equation} |{\bf e}_{F,\hat{\alpha }_3}| =\hat{\alpha }_3 \frac{c_P |\textbf {\emph {V}}|}{24 \pi }\frac{P^2_b}{P}\frac{c^2}{G M} \sin \beta, \end{equation} \tag{ 3 }$$

where c_P = −2E^grav_P/m₁c² is the gravitation self-energy fraction of the pulsar (the so-called compactness, taken to be approximately 0.21m₁; Damour & Esposito-Farèse 1992; Bell & Damour 1996), β is the unknown angle between the pulsar system's absolute velocity $\textbf {\emph {V}}$ (with respect to the reference frame of the cosmic microwave background) and the pulsar's spin vector.

We have used the above expressions and a Bayesian analysis to derive probability distributions for Δ and $\hat{\alpha }_3$ given the measured binary/pulsar parameters and additional estimates for the remaining unknown parameters. The procedure was described in detail in Stairs et al. (2005) and we summarize it here.¹⁵ For each pulsar j, we find the probability density functions (pdfs) p(|Δ||D_j, I) and $p(\hat{\alpha }_3|D_j,I)$ for probable values of Δ and $\hat{\alpha }_3$ given each pulsar's data D_j (see Table 4) and prior relevant information I.

For example, for the SEP parameter Δ we can write for each pulsar:

$$\begin{equation} p(\big |\Delta \big |,d_j|D_j,I) \propto p(D_j| \big |\Delta \big |,d_j,I) \times p(\big |\Delta \big |,d_j | I), \end{equation} \tag{ 4 }$$

where d_j represents the relevant parameters for this test, namely i, m₂, Ω, d, e_N, and θ. For these parameters, we perform Monte Carlo simulations when they are not directly measured or constrained through timing or other methods. For m₂ we use twice the range given by the P_b–m₂ relationship of Tauris & Savonije (1999). For cos i we assume a uniform distribution between 0 and 1, and combine this value with the measure mass function and m₂ to provide a value for the pulsar mass m₁ (only systems with m₁ values between 1.0 and 2.5 are kept).¹⁶ For Ω we use a uniform distribution between 0° and 360°. For d we use a Gaussian centered on the DM estimate using Cordes & Lazio (2002) and assuming an average uncertainty of 25%, or a Gaussian centered on the parallax measurement if available. The integrals over the remaining unknown parameters e_N and θ are computed separately using the measured values of e_obs and ω and the implied values of e_{F, Δ}. A pdf for the $\hat{\alpha }_3$ parameter was similarly derived; for this we also need estimates for the three-dimensional velocity of each pulsar and used Gaussian distributions in each dimension centered on the Galactic rotational velocity vector at the pulsar location with widths of 80 km s⁻¹ (Lyne et al. 1998) or, when available, we use Gaussian distributions for the proper motions to get the transverse velocities. For |Δ|, the parameter space 10⁻⁵ < |Δ| <0.1 was sampled uniformly in steps of 2 × 10⁻⁵ and for $|\hat{\alpha }_3|$ the parameter space 10$^{-22}<|\hat{\alpha }_3|<$5 × 10⁻¹⁹ was sampled uniformly in steps of 1 × 10⁻²².

Binary systems suitable for these studies are required to have large periods and small eccentricities so that additional relativistic effects are negligible. Large values of P²_b/e and P²_b/Pe have therefore been used as a general selection characteristic for choosing appropriate systems (Wex 2000; Stairs et al. 2005). In addition, the systems must be old enough and have large enough $\dot{\omega }$ so that the orientation of their orbits can be assumed to be random and that the projection of the Galactic acceleration vector on the orbit can be assumed to have been constant over the lifetime of the systems (Damour & Schäfer 1991; Wex 1997). While some pulsars might individually provide low limits for these tests, we use all 27 available systems in order to provide a more conservative upper limit that incorporates the assumptions made on the population as a whole.

Currently, a total of 27 WBMSPs are available to test the above effects and their properties are listed in Table 4. The pdfs for each pulsar are shown in Figure 7 for the SEP parameter |Δ| and in Figure 8 for the Lorentz invariance/momentum conservation parameter $|\hat{\alpha }_3|$. Since each pulsar represents an independent test of these parameters, we can multiply the individual pdfs to obtain a total pdf from our sample of pulsars.

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Using all the systems in Table 4, for |Δ| we derive a 95% upper limit of 4.6 × 10⁻³, which represents a 20% improvement from the value derived by Stairs et al. (2005). Two new pulsars are particularly constraining for this test: PSR J1711−4322 and PSR J1933−6211, which together with the improved parameters for PSR J1853+1303 have significantly contributed to the reduced upper limit for |Δ| (the secondary peak at low values of |Δ| of ∼10^−3.5 in the product pdf shown in Figure 7 is mainly due to these pulsars). For PSR J1711−4322 alone, the 95% upper limit for |Δ| is 5.6 × 10⁻⁴. Since pulsars test gravitational theories in the regime of strong fields, future improvements to the above limit are important. The fact that two pulsars discovered in the last five years were able to significantly contribute to this test is encouraging and raises the possibility that additional discoveries, and improved parameters for the objects already known (particularly PSR J1711−4322 and PSR J1933−6211), will improve the limit further.

For $|\hat{\alpha }_3|$, using the updated sample of pulsars we derive a 95% upper limit of 5.5 × 10⁻²⁰. This limit is higher than the value of 4 × 10⁻²⁰ derived by Stairs et al. (2005). We believe that the higher value better reflects the limits of this technique to constrain $|\hat{\alpha }_3|$ when a larger sample of pulsars is available. The most constraining pulsars for this test currently are PSR J1713+0747 and PSR J1853+1303 with very similar limits of 2.8 × 10⁻²⁰ and 3.1 × 10⁻²⁰ (95% confidence), respectively. The limits derived here are about 13 orders of magnitude smaller than solar system values (Will 1993) and again test the strong field limit. Further discoveries and ongoing study of present systems (particularly PSR J1853+1303) will help to place additional constraints on this test of gravitational theories.