PRECISION ASTROMETRY WITH THE VERY LONG BASELINE ARRAY: PARALLAXES AND PROPER MOTIONS FOR 14 PULSARS

An initial sample of 63 pulsars was chosen from the roughly one thousand pulsars known at the inception of this project in 2001. Pulsars were selected based primarily on flux density, declination, and dispersion measure distance estimate, so that precision astrometry was plausible using techniques refined in our previous work. A few pulsars were included since they were of particular scientific interest (e.g., the recycled pulsar J1713+0747) even though they were expected to pose technical challenges; sample completeness was not a consideration.

The field of view around each of the 63 selected pulsars was observed with the VLA between 2002 February 7 and 9 (project code AC629). Given the similar 25 m diameters of the VLA and VLBA dishes, they have well matched primary beams at the same observation frequencies. Each field was observed for ∼15 minutes at 1.4 GHz with the VLA in its highest angular resolution A-configuration (∼14 at 1.4 GHz). A spectral-line observing mode was used, with a 23 MHz spectral window or bandpass, and the correlator output was dumped every 5 s (as opposed to the more standard 10 s). The use of a spectral-line mode and short correlator dump times were both motivated by the desire to image the full field of view with minimal distortions due to time-average and bandwidth smearing (Bridle & Schwab 1999).

In order to detect potential in-beam calibrators, as well as bright confusing sources within the primary beam and beyond, a low-resolution image with a diameter of 11 was searched for sources. Small facets surrounding each source were then imaged and deconvolved at the full resolution, employing polyhedral imaging (Cornwell & Perley 1992). For each field, self-calibration was used to improve the image dynamic range, and if a substantial improvement was obtained in the dynamic range, the source-finding process was repeated in an effort to identify any weaker sources that might have been missed in the first iteration.

Each detected source was inspected for compactness and cataloged with its brightness, position, and angular separation from the pulsar. A total of 1060 sources stronger than 1 mJy were detected in the 63 fields; of these, 269 were within 25' of the target pulsar and were deemed compact enough (unresolved at 14 resolution) to warrant further investigation. At least one such candidate in-beam calibrator was found for 62 of the 63 target pulsars, and an example is shown in Figure 1. In addition, the position of each pulsar was measured to a precision of between 02 and 005, sufficient to enable the later VLBA observations.

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The angular resolution of these preliminary VLA observations was still a factor of 100 below that of the planned VLBA observations. A second set of VLA observations at 8.4 GHz was used to determine the subarcsecond structure and verify the compactness of the candidate in-beam calibrators, as well as to determine their spectral indices. Each of the 269 candidate in-beam calibrators was observed in individual pointings on 2002 April 6 with the VLA A-configuration at 8.4 GHz, which provides a resolution ∼02. Each source was observed for approximately 3 minutes in continuum mode with two 50 MHz spectral windows. Sources that were resolved were rejected, leaving 97 compact sources as potential in-beam calibrators with the VLBA, with at least one for all but eight of the original 63 pulsars. The refinement process is illustrated in Figure 1, and details of all the candidate sources are archived online.¹¹

From the original sample of 63 pulsars, 34 were selected for observations with the VLBA, most of them with good candidate in-beam calibrators. A few pulsars were judged to be strong enough and to have a cataloged VLBA calibrator that was close enough that phase referencing, in combination with wideband ionospheric calibration (Brisken et al. 2002), would provide accurate astrometry. These pulsars will be discussed in future work.

In the first epoch of VLBA observations, most of the selected pulsars were found to have at least one suitable in-beam calibrator, as illustrated in Figure 1. However, the in-beam candidates for seven pulsars were either not detected or were too heavily resolved to be useful. In addition, three pulsars were not detected in first epoch VLBA observations due to scattering or poor calibration at lower declinations; these were discarded from our observing list, and substituted with other targets.

The process outlined here was optimized to identify pulsars where a parallax measurement seemed plausible based on our previous experience. No attempt was made to preserve sample completeness or to image each possible in-beam calibrator candidate exhaustively, and as such, it is difficult to infer detailed statistical properties for the ensemble of faint, compact background sources. Of the 97 sources that were found to be compact in both 1.4 GHz and 8.4 GHz VLA A-configuration observations, the vast majority (75 sources) had a negative spectral index (S_ν ∝ ν^α; α < 0), as expected for nonthermal sources (the spectral index was determined after correcting measured flux densities at 1.4 GHz for primary beam attenuation). Specifically, α < −0.5 for 43 sources, and α < −1.0 for 16 sources. We emphasize that these results do not necessarily contradict the notion that compact sources tend to have flat spectral indices (e.g., Preston et al. 1983). Our sample is faint, and likely to include a mixture of different kinds of objects, including some core-jet sources. Additionally, our two-point spectral index might straddle a peak in the spectrum, producing an apparent distribution of spectral indices that is hard to interpret. Wall et al. (2005) and Fomalont et al. (2006) find broadly similar spectral index distributions for their source samples.

Our choice of observing frequency balances competing priorities. Radio pulsars typically have steep spectra (S_ν ∝ ν^α; α ∼ −1.6), and thus are brighter at lower frequencies. Additionally, the larger field of view at lower frequencies enhances the likelihood of finding in-beam calibrators, and tropospheric phase fluctuations are reduced at lower frequencies. On the other hand, the synthesized beam is broader, resulting in lower astrometric precision when sensitivity is the limiting factor (the case for many of the targets and calibrators here), and the phase distortions due to the differential ionosphere between the target and the calibrator increase rapidly at lower frequencies. An observing frequency around 1.5 GHz was chosen as a compromise between these effects, as demonstrated in our various pilot projects (Brisken et al. 2000; Chatterjee et al. 2001); the residual differential ionospheric effects remain the largest source of astrometric uncertainty in our program.

In an effort to minimize systematic errors between epochs, all of the observations were set up in as identical a manner as possible. Dual circular polarization was recorded simultaneously in four separate 8 MHz spectral windows. These bands were Nyquist sampled with 4-level quantization. We chose widely spread spectral windows to enable calibration of the ionosphere (Brisken et al. 2000) for cases where in-beam calibration proved difficult and the pulsar was deemed bright enough to allow this method to succeed. The in-beam calibration technique described in the present work does not make use of the wide-spread bands. The initial set of spectral windows was centered on 1414, 1489, 1594, 1684 MHz, choices that were informed by presence of known satellite-generated interference. After routinely encountering strong radio frequency interference (RFI), the second spectral window was changed to a central frequency of 1508 MHz. These spectral windows proved to be relatively RFI-free for most of the observations, although strong narrow-band RFI was occasionally observed around 1684 MHz, mainly at the Mauna Kea site. The VLBA pulse calibration system was turned off during these observations in order to reduce the total system temperature by about 3%.

At each epoch, an observation cycled between the field containing the pulsar and its in-beam calibrator(s) and a VLBI reference calibrator. The VLBI reference calibrators, which are listed in Table 1, were drawn from the standard VLBI catalogs (Beasley et al. 2002; Fomalont et al. 2003; Petrov et al. 2005) and are the ultimate means by which the observations are tied to the ICRF. The calibrator was chosen primarily for proximity to the target pulsar, though strength and compactness requirements drove some choices. The pointing direction for a target field was set to be between the pulsar and its in-beam calibrator, with the restriction that the offset from the pulsar be no more than 6'; for some pulsars, two in-beam calibrators could be identified and the pointing was set to lie between all three sources in the field. Each observation spanned a duration of 2 hr approximately centered on the transit time of the source at the center of the VLBA. We used interleaved scans of 90 s on the reference calibrator and 120 s on the target field, resulting in a total integration time of about 1 hr on the target field after accounting for slewing overheads.

We aimed to observe each pulsar for a total of 8 epochs over the course of two years, with epochs spaced approximately three months apart. Observations for a given pulsar were conducted at a similar LST at each epoch in order to provide nearly identical spatial frequency sampling ("U − V coverage"). The observations were typically scheduled within 2 weeks of parallax extrema. Between 2002 June 27 and 2005 March 25, our observing programs (BC120, BC123) accumulated a total of 278 observation blocks of 2 hr each.

A few observations failed and had to be discarded due to the proximity of the Sun to the target pulsar in the sky, a circumstance whose severity was not fully appreciated by us when the program began. Turbulence in the extended solar corona and the solar wind can cause phase fluctuations (Spangler et al. 2002) which degrade astrometry. If the target field was within ∼10° of the Sun, the images of the calibrators were severely distorted, but the solar corona and wind can affect 1.5 GHz observations of targets as far away as 45° from the Sun. In order to avoid severe phase errors, future astrometric observations should not be scheduled within about 30° of the Sun at 1.5 GHz.

A priori information about the rotational phase of a pulsar can be exploited to correlate the observations only when the pulsar is "on" and the pulse is visible, while discarding data from the remainder of each pulse period. The result is a S/N improvement scaling as ${\approx} \sqrt{P/w}$ for a pulsar with pulse period P and pulse width w, with typical improvement factor ∼3. The ephemerides used for gating the pulse signal were obtained from timing observations with the 76 m Lovell telescope of the Jodrell Bank Observatory. Typically, observing spans of a few months and overlapping the VLBA observation epoch were used to refine the pulsar rotational parameters in order to account for the effects of long-term timing noise.

The pulsar timing observations were made primarily at frequencies around 1.4 GHz, employing a dual-channel cryogenic receiver system sensitive to two orthogonal circular polarizations. The signals from each polarization were mixed to an intermediate frequency, fed through a multichannel filterbank, and digitized. The data were dedispersed in hardware and folded on-line according to the dispersion measure and topocentric period of the pulsar. The folded pulse profiles were stored for subsequent analysis.

Complete details regarding the procedure used to determine the pulsar rotational phases can be found in Hobbs et al. (2004); here we summarize only the salient aspects. Each timing observation consisted of a series of 1–3 minute integrations, with a total duration of up to ∼45 minutes. The total observing time per pulsar was determined by the amount of time required to obtain sufficient sensitivity, given the pulsar flux density. In post-processing, any of the individual integrations that appeared to be dominated by RFI were removed, the polarizations combined, and the data were averaged to produce a single total-intensity profile for the observation. Pulse times of arrival (TOAs) were subsequently determined by finding the peak in a time domain convolution of the averaged profile with a pulse profile template corresponding to the observing frequency. These TOAs were corrected to the solar system barycenter using the Jet Propulsion Laboratory DE200 solar system ephemeris (Standish 1982), analyzed using the standard TEMPO software package,¹² and transmitted to the VLBA correlator for use in the correlation. We note that our refined astrometry does not affect the pulse phase prediction from the timing solutions at a relevant level, and while gating improves the S/N ratio of the pulsar in VLBA observations, it does not otherwise influence the astrometry.

In order to mitigate time-average and bandwidth smearing (as for the VLA, Section 2.1), the correlation was performed with 32 spectral channels of 250 kHz width and then averaged for 2 s. Both time-average and bandwidth smearing should be negligible over a 30' (FWHM) field of view; however, the ionosphere can induce unpredictable delays and rates that could otherwise reduce the correlation coefficients through decorrelation for greater degrees of averaging. Only the parallel senses of polarization were correlated.

Each of the in-beam calibrators and pulsars were correlated in separate passes, with the phase reference center set in turn to the in-beam calibrator or pulsar positions. The pulsar correlation pass was gated as described above. Correlator parameters, such as clock offsets, station locations, and Earth orientation parameters, were kept identical to enable cross calibration between the correlator passes.

Corrections for various instrumental and system parameters are determined from geometric factors and ancillary data. Since they do not require observations of an astronomical source, we distinguish them as a priori calibration, following usual practice. These corrections are listed below, in the order in which they were applied.

Earth Orientation Parameters. In order to combine the signals from the various antennas, the correlator must have an accurate geometrical model of the array (Romney 1999). The model includes the orientation of the Earth in space, as described by a set of Earth orientation parameters (EOPs). The most accurate estimates of these EOPs are typically available only after a time lag of several weeks, well after data correlation. Moreover, during the course of this project, it was discovered that the VLBA correlator had inadvertently been provided with predicted rather than measured EOPs (Walker et al. 2005). The resultant differential astrometric errors are potentially as large as ∼2 mas, dwarfing the astrometric error budget. The correction for this was a delay offset computed between the delay model using the (incorrect) correlation EOPs and the actual EOPs, and this correction was applied to each data set.

Antenna Positions. The locations of the VLBA antennas have an impact on absolute astrometry at the level of ∼50 μ as per mm of position error on the longest baselines; the astrometric consequences are worse on shorter ones. The errors in differential astrometry are smaller than this by roughly the target-calibrator separation (in radians) or ∼1 μ as per mm of position error per degree of separation on the sky. On 2004 May 12, a new model for antenna positions and velocities (GSFC solution 2004b) was adopted at the VLBA correlator, introducing a typical discontinuity of ∼20 mm in antenna positions. Each data set correlated before the change was corrected by shifting antenna positions to be consistent with the 2004b solution.

Quantization Corrections. Because the VLBA uses two- or four-level quantization when sampling, the visibilities produced by the VLBA correlator differ systematically from those that would be produced by an ideal analog system. Quantization statistics were calculated for the autocorrelations and used to correct the amplitudes of the cross-correlations.

Gain and System Temperature. The measured system temperatures and antenna gains (typically ∼0.1 K Jy⁻¹ for a VLBA antenna at 1.5 GHz) were used to scale the correlation amplitudes, in the standard manner.

Parallactic Angle Correction. The VLBA antennas have altitude-azimuth mounts, so the antenna feeds rotate as the antennas track sources across the sky. Corrections were applied for the geometric phase shift introduced between the left and right circularly polarized signals due to feed rotation.

Coarse Ionospheric Correction. Precision astrometry requires correction for the differential phase and delay effects introduced by signal propagation through the Earth's ionosphere. The effects due to different ionospheric conditions above each antenna fall into two categories: image distortion and image shift. The former causes an overall decrease in the signal-to-noise of the astrometry by broadening the image. The latter introduces an unknown, unwanted position offset, directly impacting the astrometric results.

The ionospheric total electron content (TEC) can be estimated by measuring the propagation delay between Global Positioning System (GPS) signals at two different frequencies along the lines of sight to the constellation of GPS satellites. Global maps of the ionospheric TEC are routinely produced based on such measurements from around the world, and archived by the NASA Crustal Dynamics Data Information System (CDDIS¹⁵). These maps are relatively coarse, typically quantifying the ionospheric TEC on a grid with pixels spaced 5° in longitude and 25 in latitude, and updated every 2 hr. While these coarse models are far from sufficient for precision astrometry, Walker & Chatterjee (1999) use them to find a reduction in phase delay scatter by a factor of 2–5 at 2.2 GHz, and using them to correct for ionospheric dispersion delay did improve phase referencing.

Even though the spectral windows were chosen to minimize the overlap with known sources of interference, some residual RFI persisted in the data. While the usual interferometric practice is to edit data on a per-baseline basis, we concluded that RFI could be identified on an antenna basis, i.e., identified with being present only on baselines common to a specific antenna. Therefore, we could identify and flag RFI based on the autocorrelation data at each antenna, and baseline-based RFI excision was only rarely needed.

Data inspection and editing required extensive human intervention. However, since the same data were correlated multiple times, we could identify and flag RFI on the output of the first correlation pass and apply the flags to visibilities generated on all further passes.

Phase, delay, and bandpass calibrations for VLBI are typically derived from observations of calibrator sources. For a given pulsar the same bright VLBA calibrator, as listed in Table 1, was used for all three purposes, and we followed an iterative process of calibration refinement. The calibration steps listed below were all performed using AIPS.

Fringe fitting. We solved for calibrator visibility phase delays and time derivatives (rates) by fitting for a phase slope across each frequency band, assuming a point source model. The delays were smoothed by averaging over 15 minutes and interpolated for observations of the targets.

Bandpass calibration. After fringe-fitting, calibrator data from all epochs on a particular target were combined, and the inner 26 of the 32 channels in each frequency band were then imaged. Several iterations of imaging and self-calibration were performed to create a final calibrator image, averaged over all observation epochs. The resulting "global" calibrator model was used for bandpass calibration at each epoch, determining a phase and amplitude gain factor for each of the 32 channels of each frequency band, for each antenna. These corrections help to minimize baseline-dependent errors in further calibration steps.

Frequency division. The calibrator structure did not change significantly over the duration of the project, enabling us to build time-averaged models. However, in some cases the calibrator images showed significant frequency dependent structure. As such, after the initial fringe-fitting and bandpass calibration, we used separate calibrator models for each frequency band.

Phase and amplitude calibration. For each frequency band, calibrator data from all epochs were combined, and an imaging and self-calibration loop was used to derive an epoch-averaged calibrator model. The global calibrator model produced for bandpass calibration was used as the initial model in each case. Phase and amplitude calibrations were then derived independently for each epoch and frequency band, using calibrator models that were averaged over all epochs but not over frequencies. The phase and amplitude calibration was then applied to the target source (i.e., either the pulsar or the in-beam calibrator) and written to a single-source, calibrated UV database. In order to ensure that the data for the pulsar and in-beam calibrator(s) were identically calibrated, we derived the fringe-fitting, bandpass calibration, and phase/amplitude calibration for one correlation pass and copied these tables to all subsequent correlation passes.

At the end of this calibration sequence there were typically 32 calibrated single-source UV databases per pulsar and per in-beam calibrator, one for each of four frequency bands at 8 epochs.

The angular separation between an in-beam calibrator and the target pulsar (typically ∼15'–20') is much smaller than their angular separation from the phase calibrator (typically ∼2°–3°). As such, most of the residual calibration errors will be common to the pulsar and the in-beam source, and we can exploit in-beam calibrators to incrementally improve the calibration of the pulsar.

As for the phase calibrators, we created independent models for in-beam calibrators at each frequency band by combining and imaging the calibrated visibility data from all epochs at that band. A few passes of self-calibration (usually phase-only, but including amplitude self-calibration for in-beam sources brighter than ∼50 mJy) were used to improve the models. Since in-beam calibrators are usually fairly weak (∼10 mJy) and show some source structure, we required longer time averages and lower S/N cutoffs for the calibration solutions, up to a limit of 5 minutes of averaging and S/N ≳1.8. This lower S/N threshold was determined to maximize the S/N in the phase-referenced image of the pulsar after the phase transfer from the in-beam calibration solutions; calibrators with different characteristics may be optimized with slightly different cutoff values. In some cases, the in-beam source had enough resolved structure that the longest baselines of the VLBA (to Saint Croix and Mauna Kea) had to be discarded. Even in such cases, using in-beam calibration provided superior astrometric results compared to retaining the longest baselines while only referencing observations to the more distant phase calibrator.

With an epoch-averaged model for the in-beam calibrator at each frequency band, we used a final pass of phase-only self-calibration to obtain phase corrections. These corrections were applied to the calibrated pulsar data and used for subsequent imaging.

Since the target pulsars are intrinsically compact, the final imaging step is quite simple. The VLBA beam is north–south elongated, with typical dimensions of 5 × 14 mas for our chosen frequencies, observation durations, and selection of source declinations. We used pixel sizes of 1 × 1 mas and approximately natural weighting (an AIPS "robustness" parameter of 2, where 5 corresponds to pure natural weighting and −5 corresponds to uniform visibility weights; see Briggs et al. 1999). Many of the target pulsars have observed diffractive scintillation timescales at 1.5 GHz that are comparable to the 2 hr observation span, and so their sidelobes are often not well subtracted. For this reason support constraints (i.e., "boxes") were used to restrict the region to be cleaned, and we stopped the cleaning process interactively once the residuals within the support box were at the level of the image noise far from the pulsar. Finally, we determined astrometric positions for our targets at each epoch and frequency band by fitting elliptical Gaussians in the image domain.

Accurate astrometric measurements allow pulsars to be traced back through the Galaxy to potential birth sites (e.g., Hoogerwerf et al. 2000, 2001; Vlemmings et al. 2004). If a birth site can be identified, the three-dimensional birth velocity can be estimated, a kinematic age τ_kin can be inferred, and a combination of the birth spin period and braking index can be constrained. A detailed analysis of the Galactic trajectories of all pulsars with precise astrometry is beyond the scope of this work and will be presented in future (W. H. T. Vlemmings et al. 2010, in preparation), but a preliminary assessment of the current sample of pulsars is presented here.

The trajectories of the pulsars in our sample (except the old, recycled pulsar J1713+0747) are shown in Figure 6 for three different values of the (unknown) radial velocity (V_r = −200, 0 and +200 km s⁻¹). The pulsar trajectories were calculated using the three component Stäckel potential from Famaey & Dejonghe (2003) with the scaling parameters as described in Vlemmings et al. (2004). Figure 6 also shows the positions and extent of the nearby OB associations from de Zeeuw et al. (1999) and a number of open clusters compiled from the WEBDA catalog¹⁶ with a distance <5 kpc and a well defined age <10 Myr. We only plot those open clusters for which a pulsar trajectory crosses within 20 pc of the cluster center on the sky, without accounting for the distance or the motion of the cluster itself. The motion of the cluster can be ∼ several hundred pc during the lifetime of a pulsar, and needs to be taken into account in a more detailed analysis. Although a number of pulsar trajectories pass close to the OB associations or open clusters on the sky, the pulsar distance is typically incompatible with an origin in the cluster or association. However, besides the earlier identification of the likely birth site of PSR B1508+55 in one of the Cygnus OB associations (Chatterjee et al. 2005), the coarse analysis performed here reveals the possible birth location of PSR B2045 − 16 in the open cluster NGC 6604, as discussed below.

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The progenitors of NSs are massive O- and B-stars, which have a low Galactic scale height ∼63 pc, much smaller than the characteristic distance of radio pulsars from the Galactic plane. This is consistent with pulsars having significantly larger peculiar velocities than their progenitors, as recognized for example by Gott et al. (1970). As such, it is expected that most (young) pulsars should be moving away from the Galactic plane. Although Harrison et al. (1993) found that 17% of the young pulsars in their sample seemed to be moving towards the plane, later analysis indicates that after accounting for the pulsar birth scale height (z₀ = 160 ± 40 pc; Arzoumanian et al. 2002), all young pulsars do seem to be moving away from the Galactic plane (e.g., Cordes & Chernoff 1998; Brisken et al. 2003a; Hobbs et al. 2005). Any apparent motion toward the Galactic plane can often be explained by the unknown radial velocity V_r, as the velocity perpendicular to the plane V_z is given by

$$\begin{equation} V_{\rm z}=V_{\rm b}\cos {b}+V_{\rm r}\sin {b}, \end{equation} \tag{ 1 }$$

where V_b is the component of the velocity in the tangent plane along Galactic latitude b.

Alternatively, pulsars older than ∼20 Myr could potentially already be falling back towards the plane due to the effect of the Galactic potential on their trajectory. Figure 7 indicates V_z against the location with respect to the Galactic plane for the pulsars in our sample. It is apparent that for a reasonable range of V_r, all the pulsars are moving away from the Galactic plane, with the exception of PSR J0538+2817, which is located well within the pulsar birth scale height at z = −41 pc. Figure 7 also shows that, as expected, the youngest pulsars are found closest to the plane.

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Assuming that all pulsars are indeed born in the Galactic plane between −160 < z₀ < 160 pc, we can estimate the age of the pulsar from the measured z = Dsin b and V_z. The kinematic age τ_kin is given by

$$\begin{equation} \tau _{\rm kin}=(D\sin {b}-z_0)/V_{\rm z}. \end{equation} \tag{ 2 }$$

The estimated kinematic ages of the pulsars in our sample are shown in Figure 8, for −200 < V_r < +200 km s⁻¹. As deceleration in the Galactic potential will reduce V_z as pulsars get older, τ_kin will be a lower limit on the true age. Spin-down ages $\tau _{\rm sd} = P/2\dot{P}$ are also shown for comparison in Figure 8, where P is the period of the pulsar, assumed to have been born spinning rapidly and slowing steadily with a braking index n = 3. Each of these assumptions (rapid spin at birth, steady slowdown, n = 3) is known to be problematic in specific cases (e.g., Gaensler & Frail 2000; Kramer et al. 2003). However, in spite of the variety of assumptions built into both τ_sd and τ_kin, the agreement between them for most of the pulsars in our sample is remarkable.

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The only pulsars for which there is an apparent discrepancy between the two age estimates are PSRs J1713+0747, B0818 − 13, J0538+2817, and B0450 − 18. These pulsars appear to be falling towards the Galactic plane and thus have undetermined τ_kin for V_r = 0. The recycled pulsar J1713+0747 has likely already completed one or more orbits through the Galactic plane, and the simple analysis outlined above is obviously inapplicable in that case. Allowing a radial velocity slightly larger than the arbitrary cutoff of 200 km s⁻¹ resolves the discrepancy for PSR B0818 − 13. PSR J0538+2817 is still within the Galactic pulsar birth scale height, as indicated above. It is associated with the known supernova remnant S147, and the pulsar age of ∼30 kyr derived from the association (Kramer et al. 2003) is also quite inconsistent with its spin-down age (τ_sd = 620 kyr). PSR B0450 − 18 requires V_r ≳ 360 km s⁻¹ at its most probable distance inferred from bootstrap (D = 0.76 kpc), which may not be unreasonable, but in any case it has the largest uncertainties among the objects in our sample (cf. Section 5).

6.1.1. PSR B2045 − 16

Model-independent distances to pulsars provide essential calibration points for models of the Galactic electron density distribution such as NE2001 (Cordes & Lazio 2002), and one of the objectives of our observations is to use new pulsar distances to improve successors to NE2001. We have compared our astrometric distance estimates with those obtained from NE2001, as illustrated in Figure 9. In constructing NE2001, Cordes & Lazio (2002) estimated the uncertainty in the distances derived from the pulse dispersion measure (DM) to be 20%. For the purposes of our comparison, we do not include the NE2001 uncertainty estimate, but consider only whether the NE2001 distance estimate is within the 68% and 99% confidence intervals of the parallax distance.

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Of the 14 pulsars in our sample, the NE2001 distance estimates are in approximate agreement with our parallax determinations (within or just beyond the 68% confidence intervals) for 9 of the pulsars. The other five pulsars (B0031 − 07, B1508+55, B2045 − 16, B1541+09, and B0450+55) are discussed below.

Three of the discrepant pulsars ([PSR]B0031 − 07, [PSR]B1508+55, and [PSR]B2045 − 16) have high Galactic latitudes |b| > 30° and low DMs DM < 20 pc cm⁻³. The NE2001 model contains a set of low electron density components designed to reproduce the effects of local Galactic structure, namely regions such as the Local Hot Bubble (LHB), a local low density-region (LDR), and the Local Superbubble (LSB). The distances to these three pulsars are large enough that none are within any of these local, low-density regions. Thus, these local structures cannot be the explanation for the systematic discrepancies.

Gaensler et al. (2008) have re-analyzed the available DM data toward high Galactic latitude pulsars and find that the scale height of the ionized gas is approximately 1.8 kpc, about a factor of 2 higher than the thick disk component in the NE2001 model. Analyzing the DM data in combination with emission measures (EM) derived from Hα observations, they also conclude that the filling factor of the ionized gas increases with distance above the Galactic plane.

Using the Gaensler et al. (2008) re-assessment for the ionized gas scale height, we have estimated the distances to these pulsars from their DMs. In general, the resulting distances are slight over-estimates relative to their parallax distances. We conclude that the new parallax distances and the DMs of these pulsars are consistent with the conclusion of Gaensler et al. (2008) of a larger scale height for the ionized gas, though their estimated scale height of 1.8 kpc may be a bit too high. A revised fit including the new distances presented here indeed yields a slightly lower scale height (B. M. Gaensler 2008, private communication), but does not significantly alter the conclusions of Gaensler et al. (2008). We note, however, that the ISM is likely to be significantly irregular or patchy on the relevant length scales, particularly at high Galactic latitudes, where chimneys and voids are important contributors to the structure.

PSR B1541+09 is also at high Galactic latitude (b = 45°) but has a relatively large DM (35.24 pc cm⁻³). This DM is above the maximal value (≈30 pc cm⁻³) from the thick disk component in the NE2001 model, which only provides lower limits on the pulsar distance. Thus, in order to reproduce the DM, the NE2001 model includes a "clump" of enhanced electron density (contributing about 10% of the total DM), though Cordes & Lazio (2002) could find no obvious feature along the line of sight that might generate this clump (e.g., an H ii region or O star). This line of sight also passes above two spiral arms in the NE2001 model. One possibility is that the scale height of the ionized gas above one or both of these arms is larger than included in the model.

Lastly, PSR B0450+55 lies at a relatively low Galactic latitude (b = 75), with DM = 14.495 pc cm⁻³. At the parallax distance to the pulsar, the DM inferred from the NE2001 model is 33 pc cm⁻³, much larger than what is observed. The situation can plausibly be explained by a deficit of electrons in this direction. Within the NE2001 model, there are attempts to account for such deficits ("voids") in the electron density along specific lines of sight, but such modeling is limited by the sparseness of available DM-independent distance estimates.

We have examined both the Virginia Tech Spectral-Line Survey H-α image and the WHAM H-α survey (Haffner et al. 2003); neither shows any indication that the line of sight toward the pulsar has any deficit of H-α emission compared to nearby lines of sight. However, a CO survey (Dame et al. 2001) shows a tongue of CO emission that crosses the line of sight to the pulsar. While the filling factor of the CO gas along the line of sight to the pulsar may not be large, its presence suggests that not all of the line of sight may be ionized, consistent with an apparent void in the electron density along the line of sight to this pulsar.

Pulse timing provides radio pulsar positions in the Solar system reference frame, while VLBI measurements are tied to the distant quasars. Simply measuring precise positions for pulsars via the two different techniques enables fundamental reference frame ties between the Solar system and the extragalactic ICRF (e.g., Bartel et al. 1996). The recycled pulsar J1713+0747 is now one of two pulsars (along with PSR J0437 − 4715; Deller et al. 2008) with high-precision astrometry and statistically significant measurements of parallax using both pulse timing and VLBI. In Table 4 we list the astrometric parameters for PSR J1713+0747 as determined by interferometry (this work) as well as by two independent pulse timing efforts (Splaver et al. 2005; Hotan et al. 2006).

The data used by Splaver et al. (2005) span 12 years (though with a 4-year gap), allowing compensation for timing noise. In contrast, Hotan et al. (2006) used data spanning only 2.5 years, which did not allow corrections for timing noise, but allowed them to minimize systematics through the consistency of their instrumental setup. We note that the overall consistency between the timing and VLBI results for proper motion and parallax is quite good, although the position differs in declination at the ∼mas level. The longer time series allows Splaver et al. (2005) to attain higher precision in their proper motion estimates, but the uncertainties in parallax are comparable between the 12-year timing data set and the 2 year VLBI data, illustrating the complementarity of the techniques. An ensemble of such measurements on recycled pulsars has great promise for reference frame ties, as well as for the detection of gravitational waves.