THE NANOGRAV NINE-YEAR DATA SET: MASS AND GEOMETRIC MEASUREMENTS OF BINARY MILLISECOND PULSARS

The five Keplerian elements for each MSP-binary system were fitted using either the pulsar-timing binary model developed by Damour & Deruelle (1985, 1986, "DD") or the model of Lange et al. (2001, "ELL1"). Both models use ${P}_{{\rm{b}}}$ and x as timing-model parameters. The DD model is a general description of an orbit with eccentricity e and a well-defined location of periastron, such that the longitude (ω) and time (T₀) of periastron passage can be measured with numerical stability. The ELL1 model is applicable for orbits with very small eccentricities, where ω and T₀ are highly covariant. This low-e model instead parametrizes the orbit with x, ${P}_{{\rm{b}}}$, the "Laplace–Lagrange" parameters (η, κ), and time of passage through the ascending node (${T}_{\mathrm{asc}}$). Both the DD and ELL1 models contain fittable secular-variation parameters that describe PK and/or geometric phenomena.

The Keplerian elements of the NANOGrav binary MSPs, first published by Arzoumanian et al. (2015), are shown in Table 1. For NANOGrav MSPs that used the ELL1 binary timing model in the nine-year data release, we derived the DD parameters and their corresponding uncertainties using the relations derived by Lange et al. (2001) in order to show the degree to which eccentricities are well measured.

Table 1.  Keplerian Elements for the Binary MSPs in the NANOGrav Nine-year Data Release

PSR	x (lt-s)	Pb (days)	e	ω (°)	T0 (MJD)	η	κ	${T}_{\mathrm{asc}}$ (MJD)
J0023+0923	0.03484105(11)	0.13879914244(4)	0.000025(5)	82.0(12.0)	56179.082(5)	0.000024(5)	0.000003(5)	56179.08248997(8)
J0613−0200	1.0914422(5)	1.198512556680(13)	0.00000443(17)	35.0(3.0)	54890.089(10)	0.0000026(2)	0.00000362(8)	54889.991808565(12)
J1012+5307	0.5818176(6)	0.60467271380(6)	0.0000013(17)	75.0(75.0)	54901.95(13)	0.0000012(16)	0.0000003(16)	54901.95231605(11)
J1455−3330	32.3622120(3)	76.174567474(14)	0.00016965(2)	223.458(6)	55531.1454(14)	...	...	...
J1600−3053	8.8016526(10)	14.348468(3)	0.000173741(11)	181.854(16)	55419.1115(6)	...	...	...
J1614−2230	11.29119744(7)	8.68661942171(9)	0.000001333(8)	175.9(4)	55662.053(10)	0.000000096(9)	−0.000001330(7)	55658.145347857(6)
J1640+2224	55.329717(4)	175.460597(13)	0.00079725(2)	50.7313(15)	54784.4707(7)	...	...	...
J1643−1224	25.0725904(3)	147.01739554(4)	0.000505752(18)	321.849(2)	54870.5948(8)	...	...	...
J1713+0747a	32.34242188(14)	67.82513826930(19)	0.0000749402(6)	176.1963(16)	53761.0327(3)	...	...	...
J1738+0333	0.3434297(2)	0.35479073425(6)	0.0000004(10)	252.0(140.0)	55598.94(14)	−0.0000004(10)	−0.0000001(9)	55598.93613993(12)
J1741+1351	11.0033168(4)	16.3353478266(6)	0.00000998(2)	204.00(17)	55812.321(8)	−0.00000406(3)	−0.00000912(2)	55819.25468493(3)
J1853+1303	40.76952255(13)	115.653786432(6)	0.000023700(6)	346.656(11)	56128.563(3)	...	...	...
B1855+09	9.2307805(2)	12.32717119133(19)	0.00002163(2)	276.54(5)	54975.5129(19)	...	...	...
J1903+0327	105.593463(3)	95.17411738(8)	0.43667843(2)	141.6536021(15)	55776.9743424(3)	...	...	...
J1909−3744	1.89799095(4)	1.533449451246(8)	0.000000092(13)	179.0(13.0)	54514.49(6)	0.00000000(2)	−0.000000092(12)	54513.989936084(3)
J1910+1256	21.1291025(2)	58.466742058(5)	0.00023020(2)	106.013(6)	54956.3186(11)	...	...	...
J1918−0642	8.3504665(2)	10.9131775801(2)	0.000020340(18)	219.38(6)	54893.7305(17)	−0.00001291(2)	−0.000015721(13)	54897.63652454(2)
J1949+3106	7.288647(7)	1.9495344177(8)	0.0000429(3)	208.0(6)	56365.552(3)	−0.0000201(5)	−0.0000379(2)	56365.97423581(3)
B1953+29	31.4126915(2)	117.349097292(19)	0.000330230(15)	29.483(2)	55265.7096(7)	...	...	...
J2017+0603	2.1929208(9)	2.1984811364(4)	0.00000685(15)	177.0(3.0)	56201.626(15)	0.0000004(3)	−0.00000684(15)	56200.64259488(3)
J2043+1711	1.6239584(2)	1.48229078649(14)	0.00000489(13)	240.4(1.2)	56173.974(5)	−0.00000425(13)	−0.00000242(9)	56174.306240718(10)
J2145−0750	10.16410849(17)	6.83890250963(11)	0.000019295(19)	200.91(5)	54902.6174(9)	...	...	...
J2214+3000	0.0590817(3)	0.4166329463(9)	0.000009(11)	345.0(72.0)	56221.96(8)	−0.000002(10)	0.000008(11)	56221.9632381(4)
J2302+4442	51.4299676(5)	125.93529697(13)	0.000503021(17)	207.8925(18)	56302.6599(6)	...	...	...
J2317+1439	2.313943(4)	2.45933146519(2)	0.0000007(5)	101.0(42.0)	54976.1(3)	−0.0000007(5)	0.00000015(6)	54976.609358785(14)

Note. Values in parentheses denote the $1\sigma$ uncertainty in the preceding digit(s), as determined from TEMPO2. For MSPs with both DD and ELL1 parameters listed in this table, we used the ELL1 model to describe the Keplerian orbit in the TEMPO2 fit, and then derived the corresponding DD values; the 1σ uncertainties for the derived DD parameters were computed by propagating $1\sigma$ uncertainties in the fitted ELL1 parameters.

^aThe values for PSR J1713+0747 were taken from Zhu et al. (2015).

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The relativistic Shapiro timing delay (${{\rm{\Delta }}}_{{\rm{S}}}$) is incorporated into both the DD and ELL1 binary models, with the low-eccentricity expansion of ${{\rm{\Delta }}}_{{\rm{S}}}$ implemented into the latter formalism. In both timing models, ${{\rm{\Delta }}}_{{\rm{S}}}$ is a function of the "range" (r) and "shape" (s) parameters, where $s=\sin i$ in most theories of gravitation, including GR. According to GR, $r={T}_{\odot }{m}_{{\rm{c}}}$, where ${T}_{\odot }={{GM}}_{\odot }/{c}^{3}=4.925490947$ μs is a conversion factor to units of solar mass. In what follows below, we assume GR is correct in order to compute estimates of m_c from measured Shapiro-delay signals. The original analysis of the NANOGrav nine-year data set used only the traditional (${m}_{{\rm{c}}}$, $\sin i$) parameterization of the Shapiro delay, incorporating one or two Shapiro parameters if they met the F-test criteria described earlier.

For this detailed study of MSP-binary systems, we also created timing solutions that used the "orthometric" parametrization of the Shapiro timing delay (Freire & Wex 2010). The orthometric framework reparameterizes ${{\rm{\Delta }}}_{{\rm{S}}}$ as a Fourier expansion across each system's orbital period and uses two different PK parameters that are derived from the harmonics of the Shapiro-delay signal to describe the relativistic effect. In the orthometric framework, the PK parameters are either the third and fourth harmonic amplitudes of ${{\rm{\Delta }}}_{{\rm{S}}}$ (h₃ and h₄, respectively), or h₃ and the orthometric ratio $\varsigma ={h}_{4}/{h}_{3}$. The choice of (h₃, h₄) as PK parameters is most appropriate for low-e systems with $i\lt 60^\circ$, while the (h₃, ς) combination is best used for eccentric systems and low-e systems with $i\gt 60^\circ$.

While no new physical information is made available by its PK parameters, the orthometric parametrization reduces correlation between the Shapiro-delay parameters. The orthometric model therefore provides a more numerically stable solution to the timing of binary pulsars with significant Shapiro-delay signals, particularly in low-e, low $-i$ systems, where ${{\rm{\Delta }}}_{{\rm{S}}}$ is difficult to measure. The available orthometric PK parameters are related to the traditional PK parameters as nonlinear functions:

$$\begin{eqnarray}&&\varsigma =\sqrt{\displaystyle \frac{1-\cos i}{1+\cos i}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{h}_{3}=r{\varsigma }^{3}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{h}_{4}={h}_{3}\varsigma .\end{eqnarray} \tag{ 3 }$$

As shown by Freire & Wex (2010), the statistical significance of h₃ reflects the degree to which ${{\rm{\Delta }}}_{{\rm{S}}}$ is measurable and can therefore be used as a straightforward indicator for the detection of the Shapiro timing delay in a pulsar-binary system. In this study, we considered the Shapiro delay to be measurable if the estimate of h₃ was statistically significant to at least $3\sigma$. For all systems with significant ${{\rm{\Delta }}}_{{\rm{S}}}$, as well as systems with statistically significant eccentricities that did not pass the h₃ significance test, we used the $({h}_{3},\varsigma )$ parameters to describe the Shapiro timing delay. For low-e systems with no significant ${{\rm{\Delta }}}_{{\rm{S}}}$, we instead parameterized ${{\rm{\Delta }}}_{{\rm{S}}}$ using the (h₃, h₄) combination.

Given the relations between the (${m}_{{\rm{c}}}$, $\sin i$) and (h₃, ς) parameters in Equations (1) and (2), physical arguments require that ${h}_{3}\gt 0$ and $0\lt \varsigma \lt 1$. Equation (3) subsequently requires that h₄ be positive, as well. However, TEMPO2 does not impose any theoretically motivated constraints on the Shapiro-delay parameters (traditional or orthometric) during a timing-model fit; it is therefore mathematically allowed for the Shapiro-delay terms to possess values that exceed their physical limits. Such limit discrepancies are not expected to be an issue for significant ${{\rm{\Delta }}}_{{\rm{S}}}$ signals, but may occur for non-detections of the Shapiro delay due to large statistical correlation between parameters when the ${{\rm{\Delta }}}_{{\rm{S}}}$ signal is weak.

Pulsar-binary systems in tight orbits with WDs or other neutron stars typically exhibit PK effects that are observed as secular changes in the orbital elements. In order to analyze these effects, we assumed GR to be valid; we refer to the secular PK variations in this study as ${(\dot{{P}_{{\rm{b}}}})}_{\mathrm{GR}}$, ${(\dot{\omega })}_{\mathrm{GR}}$, ${(\dot{x})}_{\mathrm{GR}}$, and ${(\dot{e})}_{\mathrm{GR}}$, where the dots indicate derivatives in time. According to GR, each of the PK quantities (including the Shapiro r and s parameters) are functions of at least one of the two component masses (Damour & Taylor 1992). We neglected the ${(\dot{x})}_{\mathrm{GR}}$ and ${(\dot{e})}_{\mathrm{GR}}$ terms since these two quantities are negligible for MSP-binary systems on the timescale of the NANOGrav nine-year data set. We considered the GR terms for $\dot{{P}_{{\rm{b}}}}$, $\dot{x}$, and/or $\dot{\omega }$ in the interpretation of observed variations in PSRs J1600−3053, J1614−2230, J1903+0327, and J1909−3744.

Besides the intrinsic changes within orbits from PK effects, apparent secular variations in the orbital elements will also be induced from significant relative motion between the MSP-binary and solar-system-barycentre (SSB) reference frames (Kopeikin 1996). The secular variations in x and ω from proper motion (μ)—to which we refer in this study as ${(\dot{x})}_{\mu }$ and ${(\dot{\omega })}_{\mu }$—arise from a long-term change in certain elements of orientation as the binary system moves across the sky. The kinematic terms for $\dot{x}$ and $\dot{\omega }$ are described as trigonometric functions of i and Ω. We considered the kinematic terms for $\dot{x}$ and/or $\dot{\omega }$ in the interpretation of observed variations in PSRs J1455−3330, J1600−3053, J1640+2224, J1643−1224, J1741+1351, J1853+1303, B1855+09, J1909−3744, J1910+1256, B1953+29, J2145−0750, and J2317+1439.

A separate kinematic bias that produces observed secular variations in orbital elements can arise from several forms of relative acceleration between the MSP-binary and SSB systems (e.g., Nice & Taylor 1995), the most prominent of which are differential rotation in the Galactic disk, acceleration in the Galactic gravitational potential vertical to the disk (e.g., Kuijken & Gilmore 1989), and apparent acceleration due to significant proper motion (Shklovskii 1970). The kinematic bias from relative acceleration produces a rate of change in the Doppler shift (D) in ${P}_{{\rm{b}}}$ and ultimately produces an apparent variation in the orbital period. We refer to the component of the secular variation due to the acceleration bias as ${({\dot{P}}_{{\rm{b}}})}_{D}$, and considered this effect in the interpretation of measured variations in PSRs J1614−2230 and J1909−3744.

For sufficiently nearby pulsar-binary systems, the observed system orientation will change periodically as the Earth and the MSP orbit their respective barycenters and at their respective orbital periods. The "mixed" periodic variations in x and ω, collectively referred to as the "annual orbital parallax" (Kopeikin 1995), depend on i, Ω, and the observed parallax (ϖ) of the pulsar-binary system. Annual orbital parallax has been measured for PSRs J0437–4715 (e.g., Verbiest et al. 2008) and J1713+0747 (e.g., Zhu et al. 2015) and has been used in conjunction with the reported Shapiro timing delays and annual astrometric parallaxes to uniquely solve for the three-dimensional geometry of these two binary systems. We considered this effect when analyzing the secular variations observed in PSRs J1640+2224 and J1741+1351.

We used the procedure outlined by Splaver et al. (2002) in order to perform a statistically rigorous analysis of the 14 MSPs in the nine-year data set with significant Shapiro-delay measurements and obtain robust estimates of ${m}_{{\rm{p}}}$, ${m}_{{\rm{c}}}$, and i. For each of the fourteen MSPs, we first created a uniform, two-dimensional n × n grid of ${\chi }^{2}$ values for different combinations of ${m}_{{\rm{c}}}=r/{T}_{\odot }$ and $\cos i$, where n = 200 or greater in order to minimize artifacts from interpolation. With the exception of the noise parameters, all other timing-model parameters were allowed to vary freely when estimating the ${\chi }^{2}$ at each grid coordinate; the noise terms were held fixed at their maximum-likelihood values as determined by Arzoumanian et al. (2015). We used $\cos i$ instead of $\sin i$ as a grid coordinate since a collection of randomly oriented binary systems possesses a uniform distribution in $\cos i$.

Each ${\chi }^{2}$ map was then converted to a two-dimensional probability distribution function (PDF) by using a likelihood density of the following form,

$$\begin{eqnarray}&&p(\mathrm{data}| {m}_{{\rm{c}}},\cos i)\propto {e}^{-({\chi }^{2}-{\chi }_{0}^{2})/2}\end{eqnarray} \tag{ 4 }$$

where ${\chi }_{0}^{2}$ is the minimum value of the ${\chi }^{2}$ distribution defined on the two-dimensional grid. Bayes' theorem subsequently yields the two-dimensional posterior PDF, $p({m}_{{\rm{c}}},\cos i| \mathrm{data})$, when using the joint-uniform prior distribution of the two Shapiro-delay parameters. We then marginalized (i.e., integrated) the two-dimensional PDF over $\cos i$ to obtain the one-dimensional PDF in ${m}_{{\rm{c}}}$, and marginalized over ${m}_{{\rm{c}}}$ to obtain the one-dimensional PDF in $\cos i$. In order to obtain a PDF in ${m}_{{\rm{p}}}$, we transformed the two-dimensional (${m}_{{\rm{c}}}$, $\cos i$) probability grid to one in the (${m}_{{\rm{p}}}$, $\cos i$) space by applying the transformation rule for PDFs of random variables,

$$\begin{eqnarray}&&p({m}_{{\rm{p}}},\cos i| \mathrm{data})=p({m}_{{\rm{c}}},\cos i| \mathrm{data})\displaystyle \frac{\partial {m}_{{\rm{c}}}}{\partial {m}_{{\rm{p}}}},\end{eqnarray} \tag{ 5 }$$

where the partial derivative is evaluated by using the mass function (for a fixed value of $\cos i$).

For the two-dimensional grids, we computed ${\chi }^{2}$ values over $0\lt \cos i\lt 1$ and, unless otherwise noted, $0\lt {m}_{{\rm{c}}}\lt 1.4$ ${M}_{\odot }$. The latter upper limit approximately corresponds to the Chandrasekhar limit for a non-rotating WD. (Three exceptions to this cut-off limit are PSRs J1903+0327, J1949+3106, and J2302+4442, which are discussed individually in Section 5 below.)

We applied the same set of ${\chi }^{2}$-grid and marginalization procedures described above for the fourteen timing models with significant ${{\rm{\Delta }}}_{{\rm{S}}}$ that used the (h₃, ς) orthometric parametrization. However, we first created a ${\chi }^{2}$ grid in uniform steps of the (h₃, ς) parameters, and afterwards converted the resultant likelihood density to the (m_c, $\cos i$) probability map by using Equations (1) and (2) when applying the PDF-transformation rule.

The choice in parametrization of ${{\rm{\Delta }}}_{{\rm{S}}}$ amounts to a difference in prior probabilities on the physical parameters (${m}_{{\rm{c}}}$, $\sin i$) when performing the MCMC or ${\chi }^{2}$-grid analysis described above, due to the nonlinear relation between the physical and orthometric parameters (Equations (1)–(3)). Our first choice of prior, in (${m}_{{\rm{c}}}$, $\cos i$), is motivated by the expected distribution of randomly oriented binary systems—uniform in $\cos i$—though the choice of uniform ${m}_{{\rm{c}}}$ is arbitrary. On the other hand, Freire & Wex (2010) argue that a statistical analysis of the orthometric parameters is preferable since h₃ and ς are related to the Fourier harmonics of ${{\rm{\Delta }}}_{{\rm{S}}}$ and make no immediate assumption on the probability distributions of physical parameters. Simulations by Freire & Wex show that the one-dimensional posterior PDFs of the physical parameters will be affected in cases of low inclination, where ${{\rm{\Delta }}}_{{\rm{S}}}$ is typically weaker and the posterior density is heavily influenced by the choice of prior information. For cases in which there is a highly significant measurement of ${{\rm{\Delta }}}_{{\rm{S}}}$, such that the posterior density spans a small range of parameter space, the two choices of priors give essentially the same results. We present the results obtained from both sets of priors to demonstrate the effects such choices have on our mass measurements.

4.1.1. MCMC Analysis of Shapiro-delay Parameters

As a check on the ${\chi }^{2}$-grid procedure described above, we evaluated the parameters of each binary system using a Bayesian Markov Chain Monte Carlo (MCMC; e.g., Gregory 2005) analysis of all timing-model parameters. In the MCMC analysis, where we used the PAL2 Bayesian inference suite,²¹ the joint likelihood density includes all spin, astrometric, binary and noise terms as parameters to be sampled. The Bayesian analysis uses the traditional (${m}_{{\rm{c}}}$, $\cos i$) parameterization for the Shapiro delay, along with uniform priors on these and all other timing model parameters. We analytically marginalized the joint posterior over the DM, profile-evolution, and backend-offset parameters in order to reduce computational needs.

In principle, the MCMC analysis therefore provides a more robust exploration of the parameter space and timing-model behavior than the ${\chi }^{2}$-grid analysis, since the MCMC method samples the noise parameters, while the ${\chi }^{2}$-grid holds the noise parameters fixed. Moreover, for the MCMC analysis, the computation of ${m}_{{\rm{p}}}$ accounts for the small uncertainty in the mass function, as it uses the posterior distributions for the Shapiro-delay and Keplerian parameters.

Figure 1 shows the normalized posterior PDFs of the Shapiro-delay parameters for PSR J2043+1711 (see Section 5.13) estimated from both the ${\chi }^{2}$-grid and MCMC analyses. It is clear that the ${\chi }^{2}$-grid and MCMC analyses yield nearly identical estimates of the posterior distributions of the component masses and $\cos i$. This consistency between methods is seen for all 14 MSPs with significant ${{\rm{\Delta }}}_{{\rm{S}}}$. Thus the ${\chi }^{2}$-grid method is a reliable method for estimating posterior PDFs when using an adequate (fixed) noise model. All estimates reported below were obtained from the ${\chi }^{2}$-grid method and verified using PAL2.

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4.1.2. Constraints from ${(\dot{\omega })}_{\mathrm{GR}}$ on Shapiro-delay Parameters

4.1.3. Constraints from Geometric Variations on Shapiro-delay Parameters

A constraint on the system inclination angle can still be placed using the $\dot{x}$ measurements listed in Table 2 (e.g., Nice et al. 2001) for cases where the Shapiro timing delay is not detected. This is possible since the trigonometric term for Ω cannot exceed unity (Equation (11) in Kopeikin 1996), which corresponds to an alignment between the proper-motion vector and the projection of the orbital angular moment vector on the plane of the sky. The "magnitude" of the effect can therefore be written as $| \dot{x}{| }_{\mu ,\max }=\mu x| \cot i|$, and an upper limit on the system inclination can be calculated as

$$\begin{eqnarray}&&i\lt \arctan \left[\displaystyle \frac{x\mu }{| \dot{x}{| }_{\mathrm{obs}}}\right].\end{eqnarray} \tag{ 6 }$$

We computed a 95.4%-credible upper limit on the system inclination using Equation (6) and the 2σ lower limit of the $\dot{x}$ measurements reported in Table 4 for systems with no detected Shapiro delay.

Table 4.  Limits on Inclination from Non-detection of Shapiro Delay and/or Detection of $\dot{x}$

PSR	${i}_{\mathrm{SD}}$ (°)	${i}_{\dot{x}}$ (°)
J0023+0923	<56	...
J1012+5307	<66	...
J1455−3330	<85	<77
J1643−1224	<73	<37
J1738+0333	<70	...
J1853+1303	<74	<63
J1910+1256	<63	<63
B1953+29	<80	<77
J2145−0750	<80	<73
J2214+3000	<75	...

Note. All upper limits are at 95% confidence.

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Another constraint on the system inclination can be placed by using a non-detection of the Shapiro timing delay. The Shapiro-delay ${\chi }^{2}$ grids of pulsar-binary systems with no measurable ${{\rm{\Delta }}}_{{\rm{S}}}$ contain zero probability in regions of the (${m}_{{\rm{c}}}$, $\cos i$) space that correspond to large companion masses and high inclinations. These regions can be excluded based on statistically poor timing-model fits to the NANOGrav nine-year data sets.

However, a complication in the limit on i by using ${\chi }^{2}$ grids arises from the cut-off value in ${m}_{{\rm{c}}}$ when generating the ${\chi }^{2}$ grids as discussed in Section 4.1: the cut-off value prevents an arbitrarily large weight in probability density being assigned to values of low $\cos i$ and high ${m}_{{\rm{c}}}$, but also disregards regions of the (${m}_{{\rm{c}}}$, $\cos i$) phase space with nonzero probability density. We believe that the cut-off value in ${m}_{{\rm{c}}}$ is nonetheless justified since the only MSP with a suspected main-sequence-star companion is PSR J1903+0327. Moreover, the inclusion of more probability density would shift the upper limit on i to slightly lower values, so the upper limits we report in this study are considered to be conservative. Figure 2 shows the Bayesian-gridding and upper-limit results for PSR J0023+0923, and the upper limits for NANOGrav binary MSPs with $\dot{x}$ measurements and/or no detections of the Shapiro timing delay are provided in Table 4.

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PSR J0613−0200 is a 3.1 ms pulsar in a 1.2 day orbit that was discovered in a survey of the Galactic disk using the Parkes radio telescope (Lorimer et al. 1995). A previous long-term timing study of this MSP by Hotan et al. (2006) used the lack of a Shapiro-delay detection to place constraints on the companion mass and system inclination, such that $0.13\lt {m}_{{\rm{c}}}/{M}_{\odot }\lt 0.15$ and $59^\circ \lt i\lt 68^\circ$ if ${m}_{{\rm{p}}}=1.3{M}_{\odot }$. Two recent, independent TOA analyses of PSR J0613-0200 were performed by Reardon et al. (2016) and Desvignes et al. (2016). Reardon et al. used an 11 yr data set collected for the Parkes Pulsar Timing Array (PPTA) and did not report any secular variations or PK effects. Desvignes et al. used a 16 yr data set collected for the European Pulsar Timing Array (EPTA) to measure a significant ${\dot{P}}_{{\rm{b}}}=4.8(1.1)\times {10}^{-14}$. Neither study reports a detection of the Shapiro timing delay. A recent optical-spectroscopy study did not detect the companion to PSR J0613−0200, and placed a 5σ-detection lower limit on the photometric R-band magnitude to be $R\gt 23.8$ (Bassa et al. 2015).

For the first time, we report the detection of the Shapiro timing delay in the PSR J0613−0200 system using the NANOGrav nine-year data set. It is likely that the Shapiro-delay signal in PSR J0613−0200 went undetected by Reardon et al. (2016) and Desvignes et al. (2016) because of the better sensitivity achieved with the GBT and GUPPI backend, as reflected by the factor of two to three improvement in TOA root mean square residuals between the NANOGrav and PPTA/EPTA data sets. The ${\chi }^{2}$ grids and marginalized PDFs for PSR J0613−0200 are shown in Figure 3. Our current estimates of ${m}_{{\rm{c}}}={0.18}_{-0.07}^{+0.15}{M}_{\odot }$ and $i={68}_{-10}^{+7}$ degrees are consistent with the predictions made by Hotan et al. (2006), though our derived estimate of ${m}_{{\rm{p}}}={2.3}_{-1.1}^{+2.7}{M}_{\odot }$ is not yet precise enough to yield a meaningful constraint on the pulsar mass.

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PSR J1455−3330 is a 7.9 ms pulsar in a 76 day orbit and was discovered in a survey of the Galactic disk using the Parkes radio telescope (Lorimer et al. 1995). The long spin period of this MSP, along with its large orbit and anomalously large characteristic age, indicates potential disk instability during the transfer phase that ultimately donated little mass to the neutron star (Li et al. 1998). A recent radio-timing analysis by Desvignes et al. (2016) reported a significant $\dot{x}=-1.7(4)\times {10}^{-14}$.

We measured a significant $\dot{x}=-2.1(5)\times {10}^{-14}$ in the PSR J1455−3330 system using the NANOGrav nine-year data set. Our estimate of $\dot{x}$ is consistent with the one made by Desvignes et al. (2016) using an independent data set. We did not detect a Shapiro timing delay, as indicated by the insignificance of h₃ and the unconstrained estimate of ς listed in Table 2.

PSR J1600−3053 is a 3.6 ms pulsar in a 14.3 day orbit that was discovered in a survey of high Galactic latitudes using the Parkes radio telescope (Jacoby et al. 2007). A recent analysis of the PSR J1600−3053 system by Reardon et al. (2016) used PPTA data to make significant measurements of $\dot{x}$ and the Shapiro timing delay: ${m}_{{\rm{p}}}=2.4(1.7){M}_{\odot }$, ${m}_{{\rm{c}}}=0.34(15){M}_{\odot }$, $\sin i=0.87(6)$, and $\dot{x}=-4.2(7)\times {10}^{-15}$. Another recent and independent study by Desvignes et al. (2016) used EPTA data to measure the orthometric parameters ${h}_{3}=0.33(2)\ \mu {\rm{s}}$ and $\varsigma =0.68(5)$, consistent with the component masses and inclination measured by Reardon et al., as well as $\dot{x}=-2.8(5)\times {10}^{-15}$.

We measured a significant $\dot{\omega }$ for the first time, as well as a Shapiro timing delay in the PSR J1600-3053 system. We do not yet measure a $3\sigma$ significant $\dot{x}$, likely because the NANOGrav data span for PSR J1600−3053 is ∼6 yr, several years shorter than the EPTA and PPTA data sets. Nevertheless, we do make a tentative ∼2σ detection of $\dot{x}=-1.7(9)\times {10}^{-15}$ and have elected to include it as a free parameter in our timing solution. Our estimates of $\dot{x}$ and the orthometric parameters, ${h}_{3}=0.39(3)$ and $\varsigma =0.62(6)$, are consistent with those made by Desvignes et al. (2016).

Our measurement of $\dot{\omega }=7(2)\times {10}^{-3}\ \deg \ {\mathrm{yr}}^{-1}$ in the PSR J1600−3053 system could, in principle, be due to a combination of physical effects discussed in Section 3.3. The maximum amplitude of ($\dot{\omega }{)}_{\mu }$ for PSR J1600−3053 is ${(\dot{\omega })}_{\mu ,\max }=\mu | \csc i| \sim {10}^{-6}\,\deg \ {\mathrm{yr}}^{-1}$, which is two orders of magnitude smaller than the uncertainty level for the observed $\dot{\omega }$ in this MSP-binary listed in Table 2. Therefore, the observed $\dot{\omega }$ in the PSR J1600−3053 system cannot be due to secular variations from proper motion at the current level of precision.

The predicted GR component of $\dot{\omega }$ of PSR J1600−3053 is on the order of ${10}^{-3}\ \deg \ {\mathrm{yr}}^{-1}$ given the Keplerian parameters of the system shown in Table 1, the same order of magnitude as our measured value. We therefore used the method described in Section 4.1.2 to include both $\dot{\omega }$ and the Shapiro-delay parameters when generating the two-dimensional ${\chi }^{2}$ grid. The ${\chi }^{2}$ grids and marginalized PDFs for PSR J1600−3053 are shown in Figure 3; the constrained estimates of the component masses and inclination are ${m}_{{\rm{p}}}={2.4}_{-0.9}^{+1.5}{M}_{\odot };$ ${m}_{{\rm{c}}}={0.33}_{-0.10}^{+0.14}{M}_{\odot };$ and i = 63(5) degrees. Our constrained estimates of the Shapiro delay parameters are consistent with the estimates made by Reardon et al. (2016) and Desvignes et al. (2016).

PSR J1614−2230 is a 3.2 ms pulsar in an 8.7 day orbit with a massive WD companion; this MSP was discovered in a mid-latitude radio search of unidentified EGRET gamma-ray sources using the Parkes radio telescope (Hessels et al. 2005; Crawford et al. 2006). The PSR J1614−2230 system contains one of the most massive neutron stars known, ${m}_{{\rm{p}}}=1.97(4){M}_{\odot }$, as determined by a strategic set of observations that were made and used by Demorest et al. (2010) to measure the Shapiro timing delay in this highly inclined binary system. Demorest et al. were able to rule out nearly all models for plausible neutron-star equations of state that invoke significant amounts of exotic matter. Moreover, the PSR J1614−2230 system provided early evidence for relatively high "birth masses" of neutron stars after their formation, and before the onset of mass transfer (Tauris et al. 2011).

We made an improved measurement of the Shapiro timing delay in PSR J1614−2230 when using the NANOGrav nine-year data set, which includes a subset of the GUPPI data used by Demorest et al. (2010). The ${\chi }^{2}$ grids and marginalized PDFs for PSR J0613−0200 are shown in Figure 3. The uncertainties in both ${m}_{{\rm{c}}}=0.493(3){M}_{\odot }$ and $i=89.189(14)$ degrees have decreased such that the uncertainty in ${m}_{{\rm{p}}}=1.928(17){M}_{\odot }$ is a factor of approximately three less than that made by Demorest et al. (2010).

Although there was not a formally significant measurement of orbital decay, we nevertheless explored fitting for it. We measured ${({\dot{P}}_{{\rm{b}}})}_{\mathrm{obs}}=1.3(7)\times {10}^{-12}$. This is much larger than the component expected from general-relativistic orbital decay, ${({\dot{P}}_{{\rm{b}}})}_{\mathrm{GR}}=-0.00042\,\times \,{10}^{-12}$. Instead, it is attributable to the change in the Doppler shift due to the pulsar motion, as discussed in Section 3.3, which predicts ${({\dot{P}}_{{\rm{b}}})}_{{\rm{D}}}=1.36\,\times \,{10}^{-12}$ based on the pulsar distance and proper motion. Matthews et al. (2016) used the agreement between ${({\dot{P}}_{{\rm{b}}})}_{{\rm{D}}}$ and the observed value as a confirmation of the parallax distance to the pulsar. The precision of ${({\dot{P}}_{{\rm{b}}})}_{\mathrm{obs}}$ can be improved by extending the observing span backward using pre-GUPPI archival data published by Demorest et al. (2010) and forward (through future observations); this will eventually provide the most precise means for measuring the distance to this pulsar.

PSR J1640+2224 is a 3.1 ms pulsar in a 175 day orbit that was discovered in a Arecibo survey of high Galactic latitudes (Foster et al. 1995a, 1995b). The companion star in this system was observed using the Palomar 5.1 m optical telescope to have an effective temperature that is consistent with an old He WD (Lundgren et al. 1995). The first dedicated radio-timing study of the PSR J1640+2224 system reported a tentative detection of the Shapiro timing delay, with ${m}_{{\rm{c}}}={0.15}_{-0.05}^{+0.08}\,{M}_{\odot }$ and $\cos i={0.11}_{-0.07}^{+0.09}$ (Löhmer et al. 2005). However, Löhmer et al. did not derive a statistically significant constraint on ${m}_{{\rm{p}}}$. A subsequent TOA analysis of the NANOGrav five-year data set (Demorest et al. 2013) used Markov chain fitting methods and noted issues with the numerical stability of the observed Shapiro timing delay (Vigeland & Vallisneri 2014). The most recent radio-timing study by Desvignes et al. (2016) used EPTA data to measure a significant $\dot{x}=1.07(16)\times {10}^{-14}$, but did not measure a significant Shapiro delay.

We measured the Shapiro timing delay, $\dot{x}=1.45(10)\times {10}^{-14}$ and $\dot{\omega }=-2.8(5)\times {10}^{-4}\ \deg \ {\mathrm{yr}}^{-1}$ using the NANOGrav nine-year data set for PSR J1640+2224. The ${\chi }^{2}$ grids and marginalized PDFs of the Shapiro-delay parameters measured for this MSP are shown in Figure 4. Based on the Shapiro timing delay alone, we estimated that ${m}_{{\rm{c}}}={0.6}_{-0.2}^{+0.4}{M}_{\odot }$ and $i=60(6)$ degrees with the corresponding ${m}_{{\rm{p}}}={4.4}_{-2.0}^{+2.9}{M}_{\odot }$. The highly significant $\dot{x}$, consistent with the estimate made by Desvignes et al. (2016) at the $2\sigma$ uncertainty level, is most likely due to a secular change in the inclination of the wide binary system induced by proper motion; the current data set is not sensitive to annual orbital parallax since the annual astrometric parallax was not found to be significant for PSR J1640+2224 (Matthews et al. 2016). However, we could not reconcile the $6\sigma$-significant value of $\dot{\omega }$ with the physical mechanisms outlined in Section 3.3. In what follows below in this subsection, we explicitly discuss and reject the possibilities that were considered to explain the $\dot{\omega }$ measurement.

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The general-relativistic component of $\dot{\omega }$ cannot be the dominant term since our observed value is negative. We also rule out a significant detection of ${(\dot{\omega })}_{\mathrm{GR}}$ since, given the fitted Keplerian elements listed in Table 1, its predicted value for large assumed component masses is on the order of ${10}^{-6}\ \deg \ {\mathrm{yr}}^{-1}$. Furthermore, we reject the possibility of this measurement arising from secular orbital variations due to proper motion, since the predicted magnitude of ${(\dot{\omega })}_{\mu }$ is also on the order of ${10}^{-6}\ \deg \ {\mathrm{yr}}^{-1}$.

In principle, a nonzero value of $\dot{\omega }$ can arise from a spin-induced quadrupole term in the companion's gravitational potential due to classical spin–orbit coupling (Wex 1998); this effect has been observed in pulsar-binary systems with main-sequence companions (e.g., Wex et al. 1998), and can also be observed in pulsar-WD systems in the case where a quadrupole term is induced from rapid rotation of the WD companion. This scenario was first considered in early studies of the relativistic PSR J1141-6545 system by Kaspi et al. (2000), where they noted that classical spin-orbital coupling would cause a time derivative in the system inclination angle, di/dt, that is comparable in order of magnitude to the component of $\dot{\omega }$ due to spin–orbit coupling. We used the $\dot{x}$ measured in the PSR J1640+2224 system, the fact that $\dot{x}=d({a}_{p}\sin i)/{dt}\approx ({a}_{p}\cos i)$ di/dt, and the Shapiro-delay estimate of $\sin i$ to compute the time rate of change in the system inclination, and found that ${di}/{dt}\sim {10}^{-6}\ \deg \ {\mathrm{yr}}^{-1}$. This estimate of di/dt is two orders of magnitude smaller than the observed $\dot{\omega }$, and we therefore reject the significance of classical spin–orbit coupling in our measurement of $\dot{\omega }$ in the PSR J1640+2224 system.

While third-body effects can give rise to measurable perturbations of the pulsar-binary's Keplerian elements (e.g., Rasio 1994), such interactions with another massive component would first be observed as large variations in the pulsar-spin period. Our timing solution for PSR J1640+2224 does not show such variations in spin frequency, and so there is no evidence that J1640+2224 is a triple system. Future observations of J1640+2224, along with historical data used by Löhmer et al. (2005) and the EPTA data set, will permit for even more stringent estimates of binary-parameter variations evaluated over a larger number of orbits, and ultimately yield a more robust timing solution.

PSR J1741+1351 is a 3.7 ms pulsar in a 16.3 day orbit that was discovered in a survey of high Galactic latitudes using the Parkes radio telescope (Jacoby et al. 2007). The Shapiro delay was initially detected in this system by Freire et al. (2006).

We detected the Shapiro timing delay in the NANOGrav nine-year data set for PSR J1741+1351, as well as a highly significant measurement of $\dot{x}$ that we report for the first time. The annual orbital parallax is not significant for this MSP since the annual astrometric parallax was not significantly measured (Matthews et al. 2016). As discussed in Section 4.1.3 above, we nonetheless generated a three-dimensional ${\chi }^{2}$ grid for different values of the two Shapiro-delay parameters and Ω, in order to constrain the system geometry using both measurements. Figure 4 shows the ${\chi }^{2}$-grid results for PSR J1741+1351 when first generating a three-dimensional, uniform grid in the (${m}_{{\rm{c}}}$, $\cos i$, Ω) parameters. The two-dimensional ($\cos i$, Ω) probability grid, obtained by marginalizing over ${m}_{{\rm{c}}}$, illustrates a highly non-elliptical covariance between the two parameters. The constrained estimates of the Shapiro-delay parameters are ${m}_{{\rm{p}}}={1.87}_{-0.69}^{+1.26}{M}_{\odot }$, ${m}_{{\rm{c}}}={0.32}_{-0.09}^{+0.15}{M}_{\odot }$, $i={66}_{-6}^{+5}$ degrees, and ${\rm{\Omega }}=317(35)$ degrees.

For comparison, we over-plotted the posterior PDFs obtained from a standard two-dimensional ${\chi }^{2}$ grid over the traditional (${m}_{{\rm{c}}}$, $\cos i$) parameters, while allowing $\dot{x}$ and all other parameters to vary freely in each timing-model fit, as the gray lines in Figure 4. There are clear and significant differences between the posterior PDFs, which strongly suggest a correlation between $\dot{x}$ and one or both of the Shapiro delay parameters. The three-dimensional ${\chi }^{2}$-grid results indicate that explicit modeling of the highly significant kinematic term reduces correlation between the Shapiro-delay parameters and $\dot{x}$, and produces more sensible posterior PDFs of the component masses and system inclination that are consistent with initial results presented by Freire et al. (2006).

PSR B1855+09 is a 5.4 ms pulsar in a 12.3 day orbit with a WD companion, and is also one of the earliest MSP discoveries made using the Arecibo Observatory (Segelstein et al. 1986). This MSP-binary system was the first to yield a significant measurement of the Shapiro timing delay from pulsar-timing measurements (Ryba & Taylor 1991). The most recent long-term radio-timing study determined the pulsar mass to lie within the range of $1.4\lt {m}_{{\rm{p}}}\lt 1.8{M}_{\odot }$ (95% confidence; Nice et al. 2004). Optical follow-up observations of the companion yielded a WD-cooling timescale of ∼10 Gyr, which is twice as long as the characteristic age of the MSP (van Kerkwijk et al. 2000).

We made a highly significant measurement of the Shapiro timing delay when using the NANOGrav nine-year data set for PSR B1855+09. The ${\chi }^{2}$ grids and marginalized PDFs for PSR B1855+09 are shown in Figure 5. Our estimates of the component masses and inclination angle— ${m}_{{\rm{p}}}={1.30}_{-0.10}^{+0.11}{M}_{\odot }$, ${m}_{{\rm{c}}}={0.236}_{-0.011}^{+0.013}{M}_{\odot }$, and $i={88.0}_{-0.4}^{+0.3}$ degrees—are consistent with, and more precise than, those previously made by Kaspi et al. (1994), Nice et al. (2004), and Reardon et al. (2016).

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PSR J1903+0327 is a 2.1 ms pulsar in an eccentric, 95 day orbit with a main-sequence companion (Champion et al. 2008). This binary system, located within the Galactic disk, posed a significant challenge to the standard view of MSP formation since tidal interactions are expected to produce low-eccentricity orbits with WD companions, as is observed for all other disk MSP-binary systems. Freire et al. (2011) performed the most recent pulsar-timing analysis of PSR J1903+0327 and argued that both binary components were once members of a progenitor triple system where the main-sequence companion was in an outer orbit about an inner MSP-WD binary; this system was subsequently disrupted and produced the binary currently observed, either by a chaotic third-body interaction or full dissipation of the inner WD companion. They combined their Shapiro-delay measurement for this system with a significant measurement of $\dot{\omega }$, which they argue is due to GR, to determine the component masses and inclination with high precision: ${m}_{{\rm{p}}}=1.667(21){M}_{\odot };$ ${m}_{{\rm{c}}}=1.029(8){M}_{\odot };$ and $77.47(15)$ degrees (all 99.7% confidence). Freire et al. also measured an $\dot{x}=0.020(3)\times {10}^{-12}$ that they attributed to proper-motion bias. A recent optical analysis of radial-velocity measurements estimated the mass ratio of this system to be $q={m}_{{\rm{p}}}/{m}_{{\rm{c}}}=1.56(15)$ (68.3% confidence; Khargharia et al. 2012), consistent with the radio-timing estimate of q = 1.62(3) made by Freire et al.

We also independently measure a significant $\dot{\omega }=2.410(13)\times {10}^{-4}\ \deg \ {\mathrm{yr}}^{-1}$ in the PSR J1903+0327 system, as well as the Shapiro timing delay indicated by the significance of h₃ listed in Table 2. We do not measure a significant $\dot{x}$. The observed $\dot{\omega }$ from our data set is consistent with the measurement made by Freire et al. (2011), and so we used the methodology discussed in Section 4.1.2 to constrain the Shapiro-delay parameters assuming that GR describes the observed periastron shift. The constrained ${\chi }^{2}$ grids for PSR J1903+0327 are shown in Figure 5. From these grids, we estimated the component masses and inclination to be ${m}_{{\rm{p}}}=1.65(2){M}_{\odot };$ ${m}_{{\rm{c}}}=1.06(2){M}_{\odot };$ and $i={72}_{-3}^{+2}\ \deg \ {\mathrm{yr}}^{-1}$. The estimate of ${m}_{{\rm{p}}}$ agrees with the Freire et al. measurement at the 68.3% credibility level, while ${m}_{{\rm{c}}}$ and i are consistent at about the 95.4% credibility level. We do not adjust the uncertainty in our measurement of $\dot{\omega }$ for the maximum uncertainty in ${(\dot{\omega })}_{\mu }$, which Freire et al. do when deriving their estimates. Our derived estimate of $q=1.56(3)$ also agrees with the optical measurement and Freire et al. estimate mentioned above.

PSR J1909-3744 is a 2.9 ms pulsar in a 1.5 day orbit with a WD companion (Jacoby et al. 2005). The Shapiro timing delay has previously been observed in this system with high precision, leading to the first precise mass measurement for an MSP (Jacoby et al. 2005; Hotan et al. 2006; Verbiest 2009). Two recent, independent TOA analyses of this pulsar were performed by Reardon et al. (2016) and Desvignes et al. (2016). Reardon et al. used the PPTA data set and reported significant Shapiro-delay parameters, apparent orbital decay, and geometric variations the PSR J1909-3744 system with the following measured and derived results: ${m}_{{\rm{p}}}=1.47(3){M}_{\odot };$ ${m}_{{\rm{c}}}=0.2067(19){M}_{\odot };$ $i=93.52(9)^\circ ;$ and ${\dot{P}}_{{\rm{b}}}=0.503(6)\times {10}^{-12}$. Desvignes et al. analyzed the EPTA data set and also reported estimates of the Shapiro-delay parameters, apparent orbital decay, and geometric variations: ${m}_{{\rm{p}}}=1.54(3){M}_{\odot };$ ${m}_{{\rm{c}}}=0.213(2){M}_{\odot };$ $\sin i=0.99771(13);$ and ${\dot{P}}_{{\rm{b}}}=0.503(5)\times {10}^{-12}$.

We independently measure both Shapiro-delay parameters and ${\dot{P}}_{{\rm{b}}}$ with high significance when using the NANOGrav nine-year data set. We also make a marginal detection of $\dot{x}=-4.4(1.6)\times {10}^{-16}$ when incorporating it as a free parameter, but it does not pass the F-test criterion.

The component masses that we derived from the probability maps for J1909−3744 shown in Figure 5, ${m}_{{\rm{p}}}=1.55(3){M}_{\odot }$ and ${m}_{{\rm{c}}}=0.214(3){M}_{\odot }$, agree with the estimates made by Reardon et al. (2016) and Desvignes et al. (2016). Our estimate of $i=86.33(10)$ degrees possesses a sign ambiguity in $\cos i$, so $i=93.67(10)$ is an allowed solution for our analysis; the latter estimate agrees with the Reardon et al. and Desvignes et al. measurement.

Given our measurements of the Keplerian and Shapiro-delay parameters, the expected orbital decay in this system from quadrupole gravitational-wave emission is ${({\dot{P}}_{{\rm{b}}})}_{\mathrm{GR}}=-0.00294\,\times \,{10}^{-12}$, which is significantly less than our measurement of ${\dot{P}}_{{\rm{b}}}$. This low estimate of ${({\dot{P}}_{{\rm{b}}})}_{\mathrm{GR}}$ implies that ${\dot{P}}_{{\rm{b}}}=0.509{(9)\times {10}^{-12}\approx ({\dot{P}}_{{\rm{b}}})}_{D}$, which agrees with the measurement and assessment made by Reardon et al. (2016) and Desvignes et al. (2016). We therefore attribute the apparent orbital decay in the PSR J1909−3744 system to biases from significant acceleration between the MSP-binary and SSB reference frames. Matthews et al. (2016) used our ${\dot{P}}_{{\rm{b}}}$ measurement to find the distance to PSR J1909-3744 to be 1.11(2) kpc, in agreement with their timing-parallax distance of ${1.07}_{-0.03}^{+0.04}$ kpc.

PSR J1918−0642 is a 7.6 ms pulsar in a 10.9 day orbit with a likely WD companion that was discovered by Edwards & Bailes (2001) in a multi-beam survey of intermediate Galactic latitudes using the Parkes Radio Telescope. An optical search for the companion of PSR J1918−0642 was unsuccessful (van Kerkwijk et al. 2005), requiring that the apparent R-band magnitude of the WD be $R\gt 24$. A long-term timing study of this MSP was carried out by Janssen et al. (2010) using the Westerbork, Nançay, and Jodrell Bank radio observatories at 1400 MHz for a combined time span of 7.4 years. While only Keplerian parameters were measured, Janssen et al. (2010) combined their distance estimate to PSR J1918−0642—based on their DM estimate for this pulsar and the Cordes & Lazio (2001) electron-density model for the Galaxy—with the $R\gt 24$ limit, and the assumption that the WD cooling and pulsar spin-down are coeval, to further constrain the companion to be a He or CO WD with a thin hydrogen atmosphere. They used the mass function of the system, as well as an assumed ${m}_{{\rm{p}}}=1.35{M}_{\odot }$, to compute a minimum companion mass of ${m}_{{\rm{c}},\min }=0.24{M}_{\odot }$. A recent radio-timing analysis by Desvignes et al. (2016) used the EPTA data to measure the Shapiro delay in this system, with ${m}_{{\rm{p}}}={1.3}_{-0.4}^{+0.6}{M}_{\odot }$, ${m}_{{\rm{c}}}=0.23(7){M}_{\odot }$, and $\cos i={0.09}_{-0.04}^{+0.05}$.

We measured a highly significant Shapiro timing delay in the PSR J1918−0642 binary system using the NANOGrav nine-year data set. The probability maps computed from ${\chi }^{2}$ grids for PSR J1918−0642 are shown in Figure 6. The significance of h₃ in the PSR J1918−0642 system exceeds 27σ, a factor of approximately four better than the h₃ estimate made by Desvignes et al. (2016) when using their EPTA data set. Our precise measurements of the WD mass and inclination from the Shapiro timing delay are ${m}_{{\rm{c}}}={0.219}_{-0.011}^{+0.012}\,{M}_{\odot }$ and $i=85.0(5)$ degrees regardless of choice in the parameterization of ${{\rm{\Delta }}}_{{\rm{S}}}$. The derived estimate of the pulsar mass is the first precise estimate for this system, and is suggestive of a low-mass neutron star: ${m}_{{\rm{p}}}={1.18}_{-0.09}^{+0.10}\,{M}_{\odot }$.

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PSR J1949+3106 is a 13.1 ms pulsar in a 1.9 day orbit with a massive companion that was discovered by the ongoing PALFA survey of the Galactic plane using the Arecibo telescope (Deneva et al. 2012). The initial radio-timing study by Deneva et al. used TOAs collected with the Arecibo, Green Bank, Nançay, and Jodrell Bank telescopes over a four-year period to make a significant detection of the Shapiro timing delay in this system. They reported significant measurements of the orthometric parameters, ${h}_{3}=2.4(1)\ \mu {\rm{s}}$ and $\varsigma =0.84(2)$, as well as derived estimates of component masses and system inclination: ${m}_{{\rm{p}}}={1.47}_{-0.31}^{+0.43}{M}_{\odot };$ ${m}_{{\rm{c}}}={0.85}_{-0.11}^{+0.14}{M}_{\odot };$ and $i={79.9}_{-1.9}^{+1.6}$ degrees.

We independently measured a Shapiro timing delay in the PSR J1949+3106 using the NANOGrav nine-year data set. The probability maps computed from ${\chi }^{2}$ grids for PSR J1949+3106 are shown in Figure 6; we set ${m}_{{\rm{c}},\max }=5{M}_{\odot }$ when computing the ${\chi }^{2}$ grids since the peak-probability value is nearly equal to our usual upper limit of ${m}_{{\rm{c}},\max }=1.4{M}_{\odot }$. Our measurements of the orthometric parameters, h₃ = 2.5(5) μs and $\varsigma =0.77(10)$, are consistent with those made by (Deneva et al. 2012) at the 68.3% credibility level. The uncertainties in our measurements are comparatively larger due to the shorter time span of our data set and, therefore, less TOA coverage across the orbit. Our derived estimates of the component masses and inclination are subsequently much less stringent than those made by Deneva et al.: ${m}_{{\rm{p}}}={4.0}_{-2.5}^{+3.6}{M}_{\odot };$ ${m}_{{\rm{c}}}={2.1}_{-1.0}^{+1.6}{M}_{\odot };$ and $i={67}_{-8}^{+9}$ degrees.

PSR J2017+0603 is a 2.9 ms pulsar in a 2.2 day orbit that was initially found using the Fermi Large Area Telescope (LAT) as a gamma-ray source with no known associations; radio pulsations were discovered and subsequently timed from this source using the Nancay Radio Telescope and Jodrell Bank Observatory for nearly two years by Cognard et al. (2011). They used the mass function of the PSR J2017+0603 system, along with an assumed ${m}_{{\rm{p}}}=1.35{M}_{\odot }$, to compute a minimum companion mass of ${m}_{{\rm{c}},\min }=0.18{M}_{\odot }$.

For the first time, we detect a Shapiro timing delay in the PSR J2017+0603 system using the NANOGrav nine-year data set, with ${m}_{{\rm{c}}}={0.32}_{-0.16}^{+0.44}{M}_{\odot }$ and $i={62}_{-12}^{+9}$ degrees. The probability maps computed from ${\chi }^{2}$ grids for PSR J2017+0603 are shown in Figure 6. The observed Shapiro delay in this system is currently weak since the marginalized, one-dimensional PDF of ${m}_{{\rm{p}}}={2.4}_{-1.4}^{+3.4}{M}_{\odot }$ extends to large values of the neutron-star mass. However, we were able to make a significant detection using a comparatively small, 1.7 yr data set that includes targeted observations at select orbital phases discussed in Section 2; our measurement will improve with the inclusion of future TOAs collected at different points in the orbit.

PSR J2043+1711 is a 2.4 ms pulsar in a 1.5 day orbit that was initially found using the Fermi LAT as a gamma-ray source with no previously known associations. The radio counterpart was discovered using the Nancay and Green Bank Telescopes; the Shapiro delay was detected in this MSP-binary system using a timing model derived from TOAs collected with the Nancay, Westerbork, and Arecibo observatories over a three-year period (Guillemot et al. 2012). At the time of the initial study performed by Guillemot et al., the Shapiro timing delay was not significant enough to yield statistically meaningful estimates of the component masses and inclination angle. They placed limits on the companion mass by assuming the validity of the ${m}_{{\rm{c}}}$–${P}_{{\rm{b}}}$ relation, and derived a preferred range of $0.20\lt {m}_{{\rm{c}}}\lt 0.22{M}_{\odot };$ with this constraint, Guillemot et al. found the pulsar mass and inclination to be $1.7\lt {m}_{{\rm{p}}}\lt 2.0{M}_{\odot }$ and $i=81.3(1.0)$ degrees, respectively.

The NANOGrav nine-year data set on PSR J2043+1711, which includes the targeted Shapiro-delay observations discussed in Section 2, yields a significantly improved measurement of the component masses and system inclination as shown in Table 3; the impact of the targeted observations on the significance of ${{\rm{\Delta }}}_{{\rm{S}}}$ in the PSR J2043+1711 system was discussed by Pennucci (2015). The probability maps computed from ${\chi }^{2}$ grids for PSR J2043+1711 are shown in Figure 7. Our improved measurements of ${m}_{{\rm{c}}}={0.175}_{-0.015}^{+0.016}{M}_{\odot }$ and $i={83.2}_{-0.9}^{+0.8}$ degrees are consistent with the initial estimates made by Guillemot et al. (2012), though ${m}_{{\rm{c}}}$ is moderately lower than the range determined from the ${m}_{{\rm{c}}}$–${P}_{{\rm{b}}}$ relation. Our derived ${m}_{{\rm{p}}}={1.41}_{-0.18}^{+0.21}\,{M}_{\odot }$ is therefore slightly below the ${m}_{{\rm{p}}}$ range determined by Guillemot et al. when assuming the validity of the ${m}_{{\rm{c}}}\mbox{--}{P}_{{\rm{b}}}$ relation.

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PSR J2145−0750 is a 16 ms pulsar in a 6.8 day orbit with a WD companion and was discovered in a Parkes Telescope survey (Bailes et al. 1994). Both Phinney & Kulkarni (1994) and van den Heuvel (1994) argued that the J2145−0750 system likely experienced unstable mass transfer from "common-envelope" evolution, where the pulsar gradually expelled the outer layers of the donor, in order to explain its unusually long pulsar-spin period and massive companion compared to other binary-MSP systems. Early optical observations of the WD companion noted the difficulty in obtaining accurate photometry due to the use of a dispersion-based distance estimate and the presence of a coincident field star (Lundgren et al. 1995). However, a recent study performed by Deller et al. (2016) combined improved optical imaging with a precise VLBI distance of $d={613}_{-14}^{+16}$ pc to estimate a companion mass of ${m}_{{\rm{c}}}\approx 0.85{M}_{\odot }$. Deller et al. also detected the orbital reflex motion of J2145−0750 through their VLBI measurements, and inferred estimates of $i={21}_{-4}^{+7}$ degrees and ${\rm{\Omega }}=230(12)$ degrees.²²

We measured $\dot{x}=0.0098(19)\times {10}^{-12}$, consistent with estimates made by Reardon et al. (2016). Our estimate of ${h}_{3}=0.10(5)\mu {\rm{s}}$ does not pass the h₃-significance test, and so we do not formally measure a significant Shapiro timing delay from the radio-timing data alone. However, we used the estimate of ${m}_{{\rm{c}}}={0.83}_{-0.06}^{+0.06}\,{M}_{\odot }$ made by Deller et al. (2016) as a prior distribution when computing the posterior maps for PSR J2145−0750. The resulting constraints on $\cos i$ and ${m}_{{\rm{p}}}$ are shown in Figure 7, which yield ${m}_{{\rm{p}}}={1.3}_{-0.5}^{+0.4}\,{M}_{\odot }$ and $i={34}_{-7}^{+5}$ degrees, and are consistent with estimates made by Deller et al.

PSR J2302+4442 is a 5.2 ms pulsar in a 126 day orbit that, along with PSR J2017+0603 (Section 5.12) was initially found using the Fermi-LAT as a gamma-ray source with no known associations and observed in the radio using the Nançay Radio Telescope and Jodrell Bank Observatory for nearly two years by Cognard et al. (2011). They used the mass function of the PSR J2302+4442 system, along with an assumed ${m}_{{\rm{p}}}=1.35{M}_{\odot }$, to compute a minimum companion mass of ${m}_{{\rm{c}},\min }=0.3{M}_{\odot }$.

For the first time, we tentatively detect a Shapiro timing delay in the PSR J2302+4442 system using the NANOGrav nine-year data set. The probability maps computed from ${\chi }^{2}$ grids for PSR J2302+4442 are shown in Figure 7. Due to the weak detection of ${{\rm{\Delta }}}_{{\rm{S}}}$ and large correlation between r and s, the timing solution published by Arzoumanian et al. (2015) used a fixed value of ${m}_{{\rm{c}}}=0.355{M}_{\odot }$ that was computed from the ${m}_{{\rm{c}}}$–${P}_{{\rm{b}}}$ relation when fitting for all other timing parameters, including the Shapiro s parameter. In this study, we developed timing solutions using both the traditional and orthometric parameterizations of ${{\rm{\Delta }}}_{{\rm{S}}}$ that allowed both PK parameters to be fitted for. The value of h₃ in the PSR J2302+4442 system exceeds $5\sigma$ and therefore passes the h₃ significance test for the detection of ${{\rm{\Delta }}}_{{\rm{S}}}$.

Our estimates of the companion mass and inclination are ${m}_{{\rm{c}}}={2.3}_{-1.3}^{+1.7}{M}_{\odot }$ and $i={54}_{-7}^{+12}$ degrees, and the corresponding pulsar mass is ${m}_{{\rm{p}}}={5.3}_{-3.6}^{+3.2}{M}_{\odot }$. We computed ${\chi }^{2}$ grids with ${m}_{{\rm{c}},\max }=5{M}_{\odot }$ since the peak-probability value of ${m}_{{\rm{c}}}$ exceeds the usual upper limit of ${m}_{{\rm{c}},\max }=1.4{M}_{\odot }$. While the posterior PDFs of the component masses span a large range of mass values, the significant estimates of s and ς indicate a measurable constraint on the system inclination. The measurement of ${{\rm{\Delta }}}_{{\rm{S}}}$ will improve in significance over time since the current data set for PSR J2302+4442 only spans about 1.7 years—or ∼5 orbits, given the long ${P}_{{\rm{b}}}$ of this MSP-binary system—and so a very small fraction of the Shapiro-delay signal has been sampled. Furthermore, given the large orbit and modest inclination, we expect to see a measurable secular variation in x within the next few years.

PSR J2317+1439 is a 3.4 ms pulsar in a 2.5 day orbit that was discovered in a survey of high Galactic latitudes using the Arecibo Obsveratory and possesses one of the smallest eccentricities known (Camilo et al. 1993, 1996; Hobbs et al. 2004). The most recent radio-timing analysis of PSR J2317+1439 performed by Desvignes et al. (2016) did not yield any secular variations in orbital parameters or a significant measurement of the Shapiro timing delay when using their 17.3 year EPTA data set. However, a Bayesian-timing analysis performed by Vigeland & Vallisneri (2014) used the NANOGrav five-year data set (Demorest et al. 2013) to measure several secular variations in the binary parameters: ${\dot{P}}_{{\rm{b}}}=6.4(9)\times {10}^{-12};$ $\dot{\eta }=-2(4)\times {10}^{-15};$ and $\dot{\kappa }=2.0(7)\times {10}^{-14}$. Vigeland and Vallisneri noted that many of the posterior distributions for binary parameters of J2317+1439 changed slightly when using different priors for the astrometric timing parallax.

The original NANOGrav nine-year timing model for PSR J2317+1439 contains parameters that describe secular variations in x and the Laplace–Lagrange eccentricity parameters, with $\dot{\eta }=5.0(9)\times {10}^{-15}\ {{\rm{s}}}^{-1}$, all of which pass the F-test criterion. We found that ${\dot{P}}_{{\rm{b}}}$ did not pass the F-test, so it was not fitted in the original NANOGrav nine-year timing solution. Moreover, both the F-test and the h₃-significance test indicated that the Shapiro delay was not significant, and so we also did not initially incorporate the Shapiro-delay parameters.

Despite the statistical significance of $\dot{\eta }$, we do not believe that the PSR J2317+1439 system is experiencing physical processes that produce a changing eccentricity. For instance, if mass transfer between components were currently taking place, we would expect to observe a spin-up phase; instead, we observe seemingly "normal" spin-down properties and stable rotation that is typical of MSPs. The presence of a third massive body in a bound, hierarchical orbit about the pulsar-companion binary system would induce higher-order derivatives in spin frequency as well as additional third-body effects on the shape, size and period of the inner binary (e.g., Joshi & Rasio 1997), most of which we do not see in the NANOGrav nine-year data set. Finally, the timescale for the observed change in η is estimated to be $\eta /\dot{\eta }\approx 0.7$ years, which is implausibly short.

Because the observed $\dot{\eta }$ is physically implausible, and because covariances between it and several other parameters distort the timing solution, we chose to hold both $\dot{\eta }$ and $\dot{\kappa }$ fixed to a value of zero (i.e., no change in the eccentricity parameters of the system) while re-fitting the nine-year timing model. In this case, we found that the significance of h₃ exceeded $3\sigma$ and therefore included the Shapiro-delay parameters. We found that $\dot{x}$ did not pass the F-test, and so did not fit for it in our modified solution. The new timing model for PSR J2317+1439 fits the data well (reduced ${\chi }^{2}$ = 1.0053 for 2531 degrees of freedom), though the original model published by Arzoumanian et al. (2015) that fits for $\dot{\eta }$ and $\dot{\kappa }$ better fits the TOA data (reduced ${\chi }^{2}$ = 0.9966 for 2531 degrees of freedom).

We generated two-dimensional ${\chi }^{2}$ grids for the traditional and orthometric Shapiro-delay parameters. The probability maps and the marginalized PDFs of the component masses and system inclination are shown in Figure 8. Given the new binary timing model of PSR J2317+1439, we have made a weak detection of the Shapiro timing delay in this system since the two-dimensional probability density extends to large m_c for low inclinations, and so the system inclination angle is not as well constrained as for the other stronger detections. Our current estimates of the component masses and inclinations are ${m}_{{\rm{p}}}={4.7}_{-2.8}^{+3.4}{M}_{\odot }$, ${m}_{{\rm{c}}}={0.7}_{-0.4}^{+0.5}{M}_{\odot }$, and $i={47}_{-7}^{+10}$ degrees.

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