THE TIMING OF NINE GLOBULAR CLUSTER PULSARS

MSPs have a small intrinsic rate of spin-down ($\dot{P}_{\mathrm{int}}$) that is usually heavily contaminated by accelerations within the potential of the cluster and Galaxy (Phinney 1993). This makes it impossible to measure their spin-down related properties (surface magnetic field, spin-down luminosity, and characteristic age) directly. Instead, we calculate the limit

$$\begin{eqnarray} &&\frac{\dot{P}_{\mathrm{int}}}{P} \le \frac{\dot{P}_{\mathrm{obs}}}{P} + \frac{a_{\mathrm{c,max}}}{c} + \frac{a_{\mathrm{G}}}{c} + \frac{\mu ^2 D}{c}, \end{eqnarray} \tag{ 1 }$$

where P is the pulsar period, a_{c, max} and a_G are the accelerations due to the cluster and Galaxy, respectively, μ is the proper motion, D is the distance to the cluster, and c is the speed of light (the last term accounts for the Shklovskii effect (Shklovskii 1970)). Phinney (1992, 1993) showed that to within 10% accuracy,

$$\begin{eqnarray} &&\frac{a_{\mathrm{c,max}}}{c} \approx \frac{3 \sigma _v^2}{2 c \big(R_{\mathrm{c}}^2 + R_{\mathrm{psr}}^2\big)^{1/2}} \end{eqnarray} \tag{ 2 }$$

for R_psr < 2R_c, where σ_v is the central velocity dispersion, R_c is the core radius, and R_psr is the projected distance of the pulsar from the center of the cluster. Because all three of the clusters studied here are core collapsed, analytical (Freire et al. 2005) and numerical models are not applicable, which is why we use the approximations of Phinney. These were developed for use with pulsars in M15, which is also core collapsed. We use the values of R_c found in Harris (1996; 2010 edition). The following references are used for σ_v: Harris (1996; 2010 edition) for M62; Webbink (1985) for NGC 6544; and Valenti et al. (2011) for NGC 6624. Table 2 lists the relevant properties of each cluster. The Galactic contribution is calculated under the approximation of a spherically symmetric Galaxy with a flat rotation curve (Phinney 1993) and is

$$\begin{eqnarray} \dot{P}_{\mathrm{Gal}} &=& {-}7 \times 10^{-19} \left(\frac{P}{\rm s} \right)\nonumber\\ &&\times\, \left(\cos {b} \cos {\ell } + \frac{\delta - \cos {b} \cos {\ell }}{1 + \delta ^2 - 2 \delta \cos {b} \cos {\ell }} \right), \end{eqnarray} \tag{ 3 }$$

where b and ℓ are the Galactic latitude and longitude of the cluster, and δ = R₀/D where R₀ is the Sun's Galactocentric distance. However, the cluster term is usually dominant.

Rough mean flux density (S_ν) estimates were made by assuming that the off-pulse rms noise level was described by the radiometer equation

$$\begin{eqnarray} &&\sigma = \frac{T_{\mathrm{tot}}}{G \sqrt{n_{\mathrm{pol}} \Delta \nu \, t_{\mathrm{obs}}}}, \end{eqnarray} \tag{ 4 }$$

where T_tot is the total system temperature, G is the telescope gain, n_pol = 2 is the number of summed polarizations, and Δν is the bandwidth. For the GBT 2 GHz receiver, G = 1.9 K Jy⁻¹ and T_sys ≈ 23 K + T_sky, where T_sky is the contribution from the Galactic synchrotron emission. This was calculated by scaling the values from Haslam et al. (1982) with a spectral index of −2.6. The typical uncertainty in these estimates of S_ν is 10%–20%. We were able to record full polarization data for one observation per cluster. These were calibrated using the 25 Hz noise diode on the GBT, and we searched for a significant rotation measure (RM) by looking for a peak in the polarized flux. We searched from −1000 to 1000 rad m⁻², but in most cases, the S/N was not sufficient to constrain RM. The notable exceptions are the two pulsars in NGC 6544, for which we measure an average RM of ∼158 rad m⁻². Figure 1 shows the fully calibrated, RM corrected profiles of NGC 6544A and B.

Zoom In Zoom Out Reset image size

PSRs J1701−3006A, B, and C, and J1823−3021A and B all have timing solutions published by other authors (Possenti et al. 2003; Biggs et al. 1994). We constructed our own timing solutions based solely on our data to confirm that there were no irregularities in our data or methods. We find that nearly all our measured parameters agree with those published elsewhere to within errors.¹² The one major exception to this is PSR J1701−3006B (hereafter M62B), where we see a highly significant change in orbital period and DM compared with the results of Possenti et al. (2003). We have also measured the rate of change of the orbital period $\dot{P}_{\mathrm{b}} = -5.51(62) \times 10^{-12}$. We performed an F-test to determine if the addition of $\dot{P}_{\mathrm{b}}$ was indeed required by the data. Without it, χ² = 142.82 with 71 degrees of freedom, while after fitting for $\dot{P}_{\mathrm{b}}$ the χ² improved to 73.94 with 70 degrees of freedom. The probability that this improvement is due to chance is <1.3 × 10⁻¹¹, so the measured $\dot{P}_{\mathrm{b}}$ does indeed seem to be required by the data. M62B is known to eclipse and has an optical and X-ray counterpart (Cocozza et al. 2008). As Cocozza et al. discuss, the pulsar and companion star are almost certainly interacting. The contribution to $\dot{P}_{\mathrm{b}}$ expected from general relativity (GR) is two orders of magnitude smaller than observed, assuming M_p = 1.4 M_☉ and M_c = M_{c, min}, and there is no realistic combination of pulsar and companion masses and inclination angles that would lead to such a large relativistic $\dot{P}_{\mathrm{b}}$. As such, classic tidal effects of the extended companion are probably the cause of the change in orbital period. Our rms timing residuals are over a factor of two smaller than obtained by Possenti et al. (2003), which probably explains why we were able to detect $\dot{P}_{\mathrm{b}}$ even though we had a shorter timing baseline.

PSR J1701−3006D (hereafter M62D) is a 3.42 ms binary MSP. M62D (along with E and F) were discovered, and initial orbital solutions were given, by Chandler (2003). The orbital period is 1.1 days, with a small eccentricity,¹³ e ∼ 4.12 × 10⁻⁴. The minimum companion mass is ∼0.12 M_☉ (assuming a 1.4 M_☉ MSP) and is likely a white dwarf. We see no evidence for eclipses in M62D, but none of our observations cover conjunction, when an eclipse would be most likely. Three observations do, however, start or end within 4 hr (∼15% of the orbital period) of conjunction, and one observation ends only 15 min before conjunction. The lack of eclipses increases our confidence that the companion is a white dwarf, and not an MS star (for which eclipses should be common). We are unable to measure any precession in the longitude of periastron, so no further constraints can be placed on the mass or geometry of the system at this time. The acceleration of M62D ($\dot{P} P^{-1}$) is somewhat higher than for the majority of GC pulsars (Ransom 2008), though not exceedingly so. We note, though, that PSRs J1701−3006E and F have similarly large accelerations.

PSR J1701−3006E (hereafter M62E) is a 3.23 ms binary MSP in a circular orbit. The orbital period is only 3.80 hr, and the projected semimajor axis is only 0.0302 R_☉. When combined with the low minimum companion mass (0.031 M_☉, or only 31 Jupiter masses) and the presence of eclipses, M62E is clearly a black widow pulsar (King et al. 2005). The presence of eclipses makes it likely that the system is being seen at a high inclination, so that the true companion mass is probably close to the minimum. We have good orbital coverage of the system, and observe both ingress and egress during eclipses, which are sharp and do not lead to a significant change in DM. The eclipses are centered around orbital conjunction and occur for ∼12% of the orbit (or roughly 12 min). We see no evidence for eclipses at other orbital phases. The projected extent of the eclipsing material is ∼7 × 10⁵ km. A relationship for the radius of the companion is given by King (1988):

$$\begin{eqnarray} &&R_{\mathrm{c}} \simeq 10^4\; \hbox{km}\: (1 + X)^{5/3} \left(\frac{M_{\mathrm{c}}}{M_{\odot }} \right)^{-1/3}, \end{eqnarray} \tag{ 5 }$$

where X is the hydrogen fraction. For X = 0.7 we find R_c ∼ 8 × 10⁴ km, which is substantially smaller than the implied size of the eclipsing region. It seems likely that the eclipses are being caused by an extended region of gas surrounding the companion.

PSR J1701−3006F (hereafter M62F) is a 2.29 ms binary MSP in a circular orbit. Like M62E, it has a low minimum mass companion (∼22 Jupiter masses) and occupies a region in M_{c, min}–P_b phase space typical of non-eclipsing black widow pulsars. Indeed, despite full orbital coverage we see no evidence for flux variability as a function of orbital phase. Nonetheless, given the very low mass limit and highly circularized orbit, we still favor classifying M62F as a black widow, and the lack of eclipses probably indicates that the system is not being viewed close to edge-on (Freire 2005).

PSR J1807−2459A (NGC 6544A) has already been discussed extensively by D'Amico et al. (2001a) and Ransom et al. (2001). It is a 3.06 ms MSP in a black widow system with a highly circular orbit and M_{c, min} = 0.009 M_☉. We were able to reliably phase connect the data collected in 2009–2010 to data taken in 2011 as part of our Shapiro delay observations of PSR J1807−2500B (see below), as well as to older GBT data from 2004.¹⁴ The orbital parameters of this system were already well determined by previous authors, however no phase-coherent timing solution had ever been derived or published. Our new timing solution includes, for the first time, a precise measurement of the position of the system and its spin-down. The orbit in our solution is in good agreement with previous analyses, but significantly more precise; it allowed us to measured $\dot{P}_{\mathrm{b}}$. Like M62B, the GR contribution to $\dot{P}_{\mathrm{b}}$ is two orders of magnitude smaller than observed for a 1.4 M_☉ pulsar and minimum mass companion. However, there are no eclipses observed for NGC 6544A, so it is possible that the orbital inclination is significantly smaller than 90°, which would imply a higher companion mass and larger GR contribution. Nonetheless, for GR to contribute even 10% of the measured $\dot{P}_{\mathrm{b}}$, the companion would have to have a mass of ∼0.06 M_☉, corresponding to i = 9°. For a distribution of inclination angles that is flat in cos i, there is only a ∼1% probability of having i < 9°. It thus seems that we can safely rule out a significant contribution from GR to $\dot{P}_{\mathrm{b}}$. We can also rule out significant contamination from acceleration in the cluster because the observed $\dot{P}_{\mathrm{b}}/P_{\mathrm{b}}$ is several times larger than the maximum cluster acceleration. It thus seems likely that $\dot{P}_{\mathrm{b}}$ can be explained by normal long-term black widow behavior (Nice et al. 2000). NGC 6544A is offset from the center of the cluster by 46, or roughly 1.5 core radii. At the distance of NGC 6544, this is only 0.06 pc.

The observed $\dot{P} < 0$, which, if it were intrinsic to the pulsar, would imply that the pulsar is spinning up. In reality, $\dot{P}$ is contaminated by the acceleration of the pulsar within the gravitational field of the cluster and Galaxy, and the fact that $\dot{P} <0$ gives unambiguous evidence that the pulsar is on the far side of the cluster and accelerating toward Earth. This provides us with a probe of the mass enclosed at the projected position of the pulsar and the mass-to-light ratio (ϒ) (Phinney 1993). Following D'Amico et al. (2002),

$$\begin{eqnarray} &&\Upsilon _{\mathrm{V}} \ge 1.96 \times 10^{17}\: \frac{a_{\mathrm{c}}}{c} \left(\frac{\Sigma _{\mathrm{V}}(<\theta _{\perp })}{10^4\; {L_{\odot,V}}\: \hbox{pc}^{-2}} \right)^{-1}, \end{eqnarray} \tag{ 6 }$$

where Σ_V(< θ_⊥) is the mean surface brightness interior to the position of the pulsar. To calculate Σ_V(< θ_⊥), we assume a constant surface brightness in the core of the cluster. The cluster acceleration, a_c, is obtained by re-arranging Equation (1). In this case, the intrinsic spin-down of the pulsar is estimated by using the formula for characteristic age (τ_c)

$$\begin{eqnarray} &&\frac{\dot{P}_{\mathrm{int}}}{P} \approx \frac{1}{2 \tau _{\mathrm{c}}}, \end{eqnarray} \tag{ 7 }$$

assuming a pulsar age of 10 Gyr. We find $\dot{P}_{\mathrm{int}}/P = 1.6\times 10^{-18}\; \mathrm{s^{-1}}$. We were unable to find a proper motion for NGC 6544 in the literature, but our results are not very sensitive to this—for example, if the cluster had a transverse velocity of 100 km s⁻¹, it would affect our results at the 10% level. Combining all of this information, we find ϒ_V ⩾ 0.072 M_☉/L_☉. Massive stellar remnants in the cores of GCs should give rise to a higher ϒ_V (typically ∼3–4 M_☉/L_☉), so this result is very unconstraining.

PSR J1807−2500B (NGC 6544B; Chandler 2003) is the most intriguing system in our sample. As with NGC 6544A we were able to phase connect data spanning 2004–2011.¹⁵ The pulsar has a period of 4.19 ms and is in a highly eccentric orbit, with e = 0.747 (see Figure 3). The very high eccentricity of this system has allowed us to measure a very significant precession of periastron, $\dot{\omega } = 0.018319(12)^{\circ }\; \mathrm{yr^{-1}}$. We have also measured Shapiro delay in NGC 6544B.¹⁶ We discuss both measurements in more detail below.

4.6.1. Possible Contributions to $\mathbf {\dot{\omega }}$

The observed orbital precession could be due to any combination of tidal deformation of the companion, spin–orbit coupling, or GR effects, but for various reasons discussed below, we believe that it is due almost entirely to GR.

Tidal deformation of the companion would require an MS or giant companion. This already seems unlikely based solely on the mass function—the minimum companion mass for M_p = 1.4 M_☉ is ∼1.2 M_☉, which is well above the turnoff mass in GCs (∼0.8 M_☉). Even if we use a smaller pulsar mass (say 1.2 M_☉), the minimum companion mass is still ∼1.1 M_☉. This rules out an MS companion, unless it was a blue straggler, but given how rare these are, this seems exceedingly unlikely. Any giants would have to be close to the turnoff mass and would experience Roche lobe overflow. The resulting gas in the system would give rise to eclipses, especially if the orbit is highly inclined (and as we show below, it is) and near conjunction. As it turns out, we see no evidence for eclipses at any orbital phase, including conjunction, so a giant seems unlikely. A blue straggler would also probably cause eclipses or some other timing irregularities in the pulsar. In fact, NGC 6544B times remarkably well, with no unmodeled trends in the data and no need for an error factor to bring χ²_red = 1. Hence, we can safely rule out a significant contribution to $\dot{\omega }$ by tidal deformation.

Spin–orbit coupling is a possibility if the companion is rapidly rotating, but it scales as $|\dot{x}/x|$ times a geometrical factor, where x is the projected semimajor axis; the geometrical factor is expected to be <10 in 80% of cases (e.g., Freire et al. 2008, and references therein). Our current best limit on $\dot{x}$ implies that precession due to spin–orbit coupling should be smaller than our extremely small measurement uncertainties in $\dot{\omega }$.

For the reasons outlined above, we feel confident that the observed $\dot{\omega }$ is due almost entirely to GR. In this case the total mass of the system is given by

$$\begin{eqnarray} &&\frac{M_{\mathrm{tot}}}{M_{\odot }} = \left(\frac{\dot{\omega }}{3} \right)^{3/2} \frac{(1 - e^2)^{3/2}}{T_{\odot }} \left(\frac{P_{\mathrm{b}}}{2\pi } \right)^{5/2}, \end{eqnarray} \tag{ 8 }$$

where T_☉ = GM_☉c⁻³ and P_b is the orbital period. Using our measured values, we find M_tot = 2.56763(59) M_☉.

4.6.2. Shapiro Delay Measurement

The total system mass of NGC 6544B is fairly high for a pulsar binary system. When combined with the observed mass function, it implies a high orbital inclination and massive companion, making NGC 6544B an excellent candidate for a measurement of Shapiro delay. We observed the pulsar on ten occasions over a variety of orbital phases, including two 8 hr tracks at or near orbital conjunction.¹⁷

We used the DDH timing model developed by Freire & Wex (2010), which parameterizes the Shapiro delay as

$$\begin{eqnarray} \varsigma & \equiv & \frac{s}{1 + \sqrt{1 - s^2}} \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} h_3 & \equiv & r \varsigma ^3, \end{eqnarray} \tag{ 10 }$$

where s = sin i and r = T_☉M_c are the traditional Shapiro delay parameters. In this parameterization, h₃ quantifies the amplitude of the third harmonic and ς is the ratio of successive harmonics of the Shapiro delay. These parameters are less correlated with each other and with other orbital parameters than are s and r, and provide a better description of the combinations of orbital inclination and companion mass allowed by the timing. From the measured values of h₃ and ς we derive sin i = 0.99715(20) and M_c = 1.02(17) M_☉. On its own, the Shapiro delay does not provide precise measurements of the component masses, but the DDH model does not assume that $\dot{\omega }$ is relativistic, and thus does not make full use of the available information.

To improve our mass estimates we used the DDGR model (Taylor & Weisberg 1989), which assumes that GR correctly describes the system. In this model, M_tot and M_c are free parameters and all post-Keplerian (PK) parameters are derived from these measurements. From this we obtain M^DDGR_tot = 2.57190(73) M_☉, M^DDGR_c = 1.2064(20) M_☉, and derive M^DDGR_p = 1.3655(21) M_☉ (see Table 6 for the complete solution). This is the most precise mass ever derived for an MSP, the previous being J1903+0327 (Freire et al. 2011). To understand this exceptional precision and verify it, we created χ² maps from the DDH model, using the Keplerian parameters of the DDGR solution as a starting point. For each cos i and M_c, we calculated all five PK parameters ($\dot{\omega }$, γ, $\dot{P}_{\mathrm{b}}$, ς, and h₃) assuming they are determined solely by GR; these were kept fixed while TEMPO2 fit for all other parameters. The resulting χ² values were recoded and the associated two-dimensional (2D) probability distribution function (PDF) was calculated using the Bayesian analysis technique discussed in detail in Splaver et al. (2002). This 2D PDF was then translated into the M_c–M_p plane using the mass function of our starting DDH solution. The original 2D PDF was then collapsed onto the cos i and M_c axes, with the derived 2D PDF collapsed into the M_p axis (see red lines in Figure 5), generating 1D PDFs for the latter quantities. These are slightly asymmetric; they have medians and ±1σ percentiles at cos i^med = 0.097^+0.017_−0.015, M^med_p = 1.3649^+0.0017_−0.0022 M_☉, and M^med_c = 1.2068^+0.0022_−0.0016 M_☉. The maximum probability occurs at cos i^max = 0.095, M^max_p = 1.3654 M_☉, and M^max_c = 1.2063 M_☉. These are in excellent agreement with the best-fit results of the DDGR model, and confirm the value and the small uncertainty of the masses.

Zoom In Zoom Out Reset image size

The regions of the cos i–M_p plane (and derived M_p–M_c plane) allowed by each measured PK parameter are depicted in Figure 5. This allows us to understand the reason for such a precise mass: the constraint provided by $\dot{\omega }$ (the total mass M_tot, according to GR) cuts the cos i–M_c (or M_p–M_c) regions allowed by the Shapiro delay very sharply.

The measured companion mass strengthens the case made in Section 4.6.1 that the companion must be a massive white dwarf or second neutron star. The companion is the most massive known around a fully recycled pulsar. If this is a double neutron star system, it is only the second known in GCs, the first being M15C (Anderson et al. 1990; Jacoby et al. 2006). It is likely that the pulsar was recycled by a different companion, which was then ejected in an exchange interaction. Deep Hubble Space Telescope images of the cluster may be able to detect a white dwarf companion, while a non-detection of the companion would strengthen the case for a double neutron star system.

We have measured three PK parameters ($\dot{\omega }$, ς, and h₃). Assuming GR is correct, all three agree on the same region of the M_c–M_p plane. As such, GR passes this test, albeit at relatively low precision. Our current best measurement of a fourth PK parameter, the gravitational redshift, is γ = 0.026(14) s. This is consistent with the GR prediction of 0.014 s. We are continuing to monitor this system, and in a few years expect to have a more precise measurement of γ that will allow for a second test of GR.

PSR J1823−3021C (NGC 6624C) has P = 0.406 s, unusually slow among GC pulsars but, surprisingly, the second such slow pulsar in NGC 6624. It was discovered by Chandler (2003); we have now measured, for the first time, $\dot{P} = 2.25 \times 10^{-16}\; {\rm s}\: \mathrm{s^{-1}}$. Using Equation (2), we find $\dot{P}_{\mathrm{c,max}} = 1.7 \times 10^{-17}\; {\rm s}\: \mathrm{s^{-1}}$, over an order of magnitude smaller than the measured $\dot{P}$. The contributions from the Galaxy and the Shklovskii effect are even smaller still, so it is clear that the observed $\dot{P}$ is due almost entirely to the intrinsic pulsar spin-down. The implied characteristic age and surface magnetic field is τ_c ∼ 2.8 × 10⁷ yr and 3.1 × 10¹¹ G. This makes NGC 6624C, like NGC 6624B, similar to the "normal," NRPs usually seen in the Galactic disk. As explained in Section 1, NRPs are typically assumed to form in core-collapse supernova, which require massive stars that have not existed in GCs for billions of years. Since NRPs have lifetimes ≪10⁹ yr, core-collapse supernovae cannot explain the presence of NGC 6624C and pulsars like it. The leading alternative explanation is ECS. The kick velocities (v_kick) that pulsars receive when they form via ECS are not well known, though there is evidence that v_kick ∼ 10 km s⁻¹ (Pfahl et al. 2002; Kitaura et al. 2006; Dessart et al. 2006; Martin et al. 2009; Wong et al. 2010). The escape velocity from the center of NGC 6624 is ∼35 km s⁻¹, which effectively places an upper bound on the v_kick that NGC 6624C received. This supports the notion that ECS kicks are much smaller than those induced by core-collapse supernovae.

GC NRPs can also impose useful statistical constraints on the properties of ECS (Boyles et al. 2011; Lynch et al. 2012, in preparation). NGC 6624C is one of the only four GC NRPs known, making it an important addition to this rare family of pulsars.

As part of this work, we discovered three previously unknown MSPs, J1823-3021D, E, and F (hereafter NGC 6624D, E, and F). However, we were unable to obtain full timing solutions for them. This was due to the very low S/N of the detections of all three pulsars, due in part to their low flux densities, but also due to a data processing error on our part. As is typical, we de-dispersed the raw data and combined many frequency channels to reduce data volume (i.e., sub-banding), discarding the raw data afterward. However, we accidentally de-dispersed at the wrong DM for many of our observations, thereby adding significant dispersive smearing and degrading the signal from these already weak pulsars even further. Without the raw data, we are unable to remedy this error. PSR J1823−3021A was bright enough that it could still be detected, and the dispersive smearing was not large enough to significantly impact the two long-period pulsars. Despite this, we could still determine some of the basic properties of NGC 6624D, E, and F.

We are able to confirm that NGC 6624D and E are isolated MSPs with P = 3.02 and 4.39 ms, respectively. NGC 6624D was detected on many occasions, but the TOA errors were quite large. We were able to measure the position with some accuracy, although without a reliable measurement of $\dot{P}$, we are unable to break all the covariances in the fit, so these coordinates should be used cautiously. NGC 6624E was only detected in three observations but there was no variation in the apparent period of the pulsar between these observations, nor was there any evidence of acceleration. Both of these effects are expected if the pulsar were in a binary system. We were unable to phase connect any of our observations due to large relative errors on the measured period of the pulsar, and we cannot constrain its position or $\dot{P}$.

NGC 6624F is an eclipsing binary MSP with P = 4.85 ms. Like NGC 6624E, NGC 6624F was only detected three times. In each case, we were able to clearly measure variations in P, $\dot{P}$, and $\ddot{P}$ arising from orbital motion. As it turns out, the apparent period of the pulsar was similar and the acceleration was positive in all three observations, indicating that the observations occurred at overlapping orbital phases near inferior conjunction. Because the pulsar had the exact same apparent period at some point during each of our observations, there must have been an integer number of orbits between these times. We calculated the exact moment during each observation when the pulsar had a reference barycentric period of precisely 4.85 ms through a Taylor expansion of P, $\dot{P}$, and $\ddot{P}$, and a bisection method. An orbital period of 0.8827 days implies exactly 106.0003 and 146.9996 complete orbits between our three observations. The fact that we were able to fit such a precise integer number of orbits gives us confidence that this is the true orbital period of the system. If NGC 6624F is in a circular orbit, each observation would mark points on an ellipse in the P–$\dot{P}$ plane (Freire et al. 2001). With three observations and a good determination of the spin and orbital period, we can, in principle, uniquely constrain the equation of this ellipse, which in turn can provide the intrinsic P, P_b, and asin i/c. We attempted to perform this fit to our data using a least-squares minimization routine and 0.01 lt-s ⩽ asin i/c ⩽ 1000 lt-s. We found a best-fit asin i/c ≈ 4.4 lt-s, but this solution has poor predictive power (i.e., when we folded our data using this solution, the pulsar was not detected). We were also unable to construct a coherent timing solution in TEMPO2 using this starting orbital solution. We believe that the lack of coverage at multiple orbital phases is limiting our ability to accurately determine asin i/c. It is also possible that NGC 6624F is not in a circular orbit.