RADIO AND γ-RAY CONSTRAINTS ON THE EMISSION GEOMETRY AND BIRTHPLACE OF PSR J2043+2740

The rotational ephemeris for PSR J2043 + 2740, which is required to fold the γ-ray photons with the pulsar's period, was obtained from regular timing observations with the Lovell 76 m radio telescope, at 1.4 GHz. The Jodrell Bank radio observatory has been timing this pulsar regularly since 1996, as part of its extensive pulsar-timing program (Hobbs et al. 2004). In our analysis, we used timing data that spanned from 2008 June 17 to 2009 October 17, covering 78, 12 minute integrations of the radio pulses, from each of which a precise time of arrival (TOA) was determined. The set of TOAs was analysed with the pulsar-timing package TEMPO2 (Hobbs et al. 2006), which we used in order to fit a timing model to the TOAs, describing the pulsar rotation during the above range of dates. The only free parameters of the timing model were the pulsar spin-frequency and its first two time derivatives, while the pulsar position and DM were deemed sufficiently well-determined from previous observations. After fitting, the pulsar ephemeris was a good description of the pulsar rotation, with the rms of the timing residuals being ≈120 μs across the entire fitted range. The uncertainty in DM results in a possible error in the time delay between the radio and γ-ray frequencies (with f_LAT → ∞), via the dispersion law, δt = (4.15 × 10⁶ ms) × σ_DM/f². Substituting for the DM uncertainty and the radio frequency gives δt ≈ 0.21 ms, which corresponds to only 0.2% phase error; this is of the same order of magnitude as the uncertainty of the ephemeris. However, given the low number of γ-ray events collected from PSR J2043 + 2740, so far, the impact of both errors on the pulsar's γ-ray light curve is negligible.

The timing parameters used in this study will be made available on the servers of the Fermi Science Support Center (FSSC).⁵⁷

In addition to the timing observations, we performed polarization observations of PSR J2043 + 2740 at 1.4 GHz, with the Effelsberg 100 m radio telescope. The 21 cm receiver of the Effelsberg observatory is equipped with left and right circular polarization feeds, with a system equivalent flux density of ≈20 Jy. The two polarization channels were fed into a Digital Filterbank back end, which was set to de-disperse the 150 MHz of available bandwidth, split into 1024 frequency channels. For calibration purposes, during the entire 45 minute observation we triggered the receiver's noise diode at the pulsar frequency but separated by 0.5 periods from the pulse position. The noise diode injects an artificial, pulsed signal that is 100% linearly polarized, and which can be used as a reference source for correcting for the gain differences between the polarization feeds, in the post-processing.

We used the PSRCHIVE software package (Hotan et al. 2004) to calibrate the Effelsberg data and produce a high signal-to-noise (S/N) polarization profile of PSR J2043 + 2740. The total-intensity, linear-, and circular-polarization profiles of this pulsar are shown in Figure 1. Furthermore, in the top panel of the same figure we show the polarization position angle (hereafter P.A.) profile across the pulse, given by , where Q and U are the Stokes parameters of the linearly polarized emission at each pulse longitude: only P.A.s with S/N>4 are shown in the plot.

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Adopting a standard rotating vector model (RVM), described in Radhakrishnan & Cooke (1969), we determined the model-parameter space from fits to the observed P.A.s across the pulse: i.e., α, the inclination angle between the magnetic and rotating axes; ζ = α + β, the viewing angle, where β is the minimum angular separation between the observer's line of sight and the magnetic axis, which occurs at rotational phase ϕ₀; and P.A.₀, the P.A. at ϕ₀. As is the case with many other polarized pulsars, the P.A. profile of PSR J2043 + 2740 reveals to us only part of the full swing of the magnetic axis across our field of view. Therefore, there are large uncertainties on the fitted parameters. In order to estimate the confidence intervals of α and ζ, we fitted the P.A.s with an RVM model: we used a 500 × 500 grid in α = {0°–180°} and ζ = {0°–180°} and left ϕ₀ and ψ₀ as free parameters to be fitted with a χ²-minimization procedure. Each grid point was then assigned to the calculated χ² value from the respective fit and 1σ, 3σ, and 5σ contours were generated.

Despite the aforementioned polarization calibration procedure, which largely corrects for the parallactic angle changes during the observation and the frequency-dependent gain differences between the dipoles (for a description see Johnston 2002), the P.A.s that we measured with our back end are subjected to additional instrumental rotations by the different components between the receiver and the back end. The instrumental rotation is assumed to be time and frequency independent, and one can account and correct for this constant shift in the P.A.s by observing with the same setup a set of highly polarized pulsars with well-known polarization properties. Given that these pulsars are also calibrated, the difference between the measured P.A.s (corrected to infinite frequency) and previously published absolute P.A. values should correspond to the instrumental rotation. Application of the measured instrumental rotation to the calibrated profile of any pulsar observed with the same system should result in absolute P.A.s.

We performed polarization observations of five pulsars—PSR B0355 + 54, PSR B0740 − 28, PSR B1929 + 10, PSR B1953 + 50, and PSR B2154 + 40—with the same receiver setup and produced calibrated P.A. profiles across the pulse. Using the calibrated P.A.s, we measured the rotation measures (RMs) of these pulsars by fitting for the P.A. rotation across our 150 MHz band. Following that, we used the resulting RMs to correct the measured P.A.s to infinite frequency. Then, we calculated the mean difference between the published values of P.A.₀ and those from our measurements in our sample of five pulsars. Absolute polarization profiles for these pulsars were found in Johnston et al. (2005) and Carr (2007). The calculated ΔP.A. between the measured and published P.A.s was P.A._meas − P.A._pub = 136 ± 28.

For PSR J2043 + 2740, following the above procedure, we measured RM = −92.7 ± 1.9 rad m⁻² and corrected the P.A.s to infinite frequency, according to that value. Finally, by subtracting ΔP.A. from all measured P.A. values, we calculated the absolute polarization-angle profile shown in the top panel of Figure 1. The RVM-model contours from the radio-polarization data allowed us to place constraints on the ranges of the α and ζ parameters (see Section 4.1). Following the aforementioned RVM-fitting approach but, instead, using a grid in ϕ₀ and P.A.₀ and fitting for α and ζ, we were able to generate confidence contours for ϕ₀ and P.A.₀ (Figure 2). The best solution (χ²_r ≈ 0.8) gave ϕ₀ = 1.032 ± 0.002 and P.A.₀ = 17° ± 6°.

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The raw event data from the Fermi LAT observations were subjected to cuts, in order to reduce the γ-ray background. We kept the “diffuse” class events as defined under the P6_V3 instrument response function (IRF), having the best probability of being γ-ray photons (Atwood et al 2009), and rejected all events with zenith angles greater than 105° in order to reduce the contribution of the Earth's γ-ray albedo.⁵⁸ We kept events within an energy-dependent radius corresponding to 68% of the point-spread function (PSF) of the LAT, from the position of J2043 + 2740 (α_J2000 = 310931, δ_J2000 = 27682), θ₆₈ = 08 × E^−0.8 with E in GeV, with a minimum value of 01 for E>10 GeV. A maximum radius of 1° around the pulsar was imposed to remove as many low-energy γ rays due to the Galactic diffuse emission as possible. The remaining 1244 photons were phase-folded using the radio ephemerides described above, and the Fermi plug-in provided by the LAT team and distributed in the TEMPO2 pulsar timing package (Hobbs et al. 2006).

Figure 3 shows the generated light curves for all events above 0.1 GeV, those between 0.1 and 0.3 GeV, 0.3 and 1 GeV, and all events above 1 GeV. The highest-energy event in the data after the application of cuts was 4.9 GeV, and falls at 0.30 in phase. The bottom plot, in the same figure, shows the radio profile of PSR J2043 + 2740 from Jodrell Bank observations at 1.4 GHz. All profiles were phase-aligned to the maximum of the radio emission (set by definition to ϕ = 0), to allow a comparison study between the longitude of radio and high-energy emission. The overall light curve above 0.1 GeV shows two significant peaks of γ-ray emission, which is in agreement with previous Fermi observations (Abdo et al. 2010a), but departs from the marginally significant light curve recorded by AGILE (Pellizzoni et al. 2009). Application of the binning-independent H-test for uniformity (de Jager et al. 1989) to those events yielded H = 69.5, which translates to a ≈7σ detection (de Jager & Büsching 2010). Inspection of the remaining histograms showed that the phase region defining P1 in the aggregate light curve is mainly populated by γ rays coming from the lower-energy windows, i.e., 0.1–1 GeV. In contrast, the γ rays that form P2 in the total light curve mainly come from the highest-energy window, i.e., >1 GeV. We therefore conclude that there is a spectral dependence of the γ-ray light curve, with the second peak harder than the first one.

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In order to accurately determine the peak positions in the light curve of PSR J2043 + 2740, a C-test (de Jager 1994) was applied. This test, unlike the H-test, is sensitive to phase and pulse-width information, which allows one to maximize the significance of the signal by scanning in phase and pulse width around the roughly known pulse positions. A grid of 200 × 60 trials of peak position, ϕ, and pulse width, W, respectively, was generated and a probability of random occurrence was calculated for each point on the grid. The pulse width was defined as the full width at half-maximum (FWHM) of a Gaussian pulse shape. From the test, we found that ϕ_P1 = 0.20 ± 0.03 and W_P1 = 0.19 ± 0.06, and ϕ_P2 = 0.55 ± 0.02 and W_P2 = 0.06 ± 0.01 describe well the peak position and width.

From the above values we derive a phase lag between the maximum of the radio emission peak and the first γ-ray peak of δ = 0.20 ± 0.03, and a separation between the two γ-ray peaks of Δ = 0.35 ± 0.04, in agreement with previously released results (Abdo et al. 2010a).

The phase-averaged spectrum for PSR J2043 + 2740 was obtained by performing a maximum likelihood analysis (Mattox et al. 1996) of the LAT data within 15° from the pulsar, using the standard tool “gtlike.” The Galactic diffuse emission was modeled using the gll_iem_ v02 map cube. The extragalactic diffuse and residual instrument background components were modeled jointly using the isotropic_iem_v02 template. Both models are available for download with the Fermi Science Tools.⁵⁹ All sources within 15° from J2043 + 2740 in the 11 month catalog of point sources were included in the analysis, and parameters for sources more than 5° away from the pulsar were fixed in the fit (Abdo et al. 2010c). We modeled the emission from J2043 + 2740 as a simple power law and an exponentially cutoff power law of the following form:

In this expression, Γ is the power-law index, E_c is the cutoff energy, and N₀ is a prefactor term. The exponentially cutoff power-law emission model is preferred at the 5.5σ level over the simple power-law model. We obtain N₀ = (1.2 ± 0.2) × 10⁻⁸ photons cm⁻² s⁻¹ GeV⁻¹, Γ = 1.1 ± 0.2, and E_c = 0.8 ± 0.1 GeV. We also modeled the pulsar with a super-exponentially cutoff power law of the form N₀(E/1 GeV)^−Γexp [ − (E/E_c)^β], where β was left free in the fit. We measured β = 1.0 ± 0.4, and therefore we conclude that the simple exponential cutoff power-law models the present data well.

Integrating Equation (1) for energies above 0.1 GeV gives an integral photon flux F_>0.1 GeV = (2.2 ± 0.4) × 10⁻⁸ photons cm⁻² s⁻¹, and an energy flux G_>0.1 GeV = (1.5 ± 0.2) × 10⁻¹¹ erg cm⁻² s⁻¹. The quoted errors are statistical only. The effect of the systematic uncertainties in the effective area on the spectral parameters is δΓ = (+0.3, − 0.1), δE_c = (+20%, − 10%), δF_>0.1 GeV = (+30%, − 10%), and δG_>0.1 GeV = (+20%, − 10%) (Abdo et al. 2010a). These systematic uncertainties were assessed by making the analysis as described above, but using an effective area for the LAT modified by ±10% at 0.1 GeV, ±5% at 0.5 GeV, and ±20% at 10 GeV with linear extrapolations in log space between. Our values for Γ, E_c, F_>0.1 GeV, and G_>0.1 GeV are in good agreement with those of Abdo et al. (2010a), but with improved statistical uncertainties.

All recent detections by Fermi suggest that the observed γ-ray emission from pulsars is distributed over a large fraction of the celestial sphere, i.e., f_Ω ≳ 1, where f_Ω is a flux-correction factor for the fact that the actual phase-averaged flux of a pulsar, integrated over the whole celestial sphere, may be higher (f_Ω>1) or lower (f_Ω < 1) than would be inferred from assuming isotropic emission based on the phase-averaged flux for the Earth line of sight (Watters et al. 2009).

The phase-averaged flux integrated over the whole sky is a measure of the size of the emission region in the pulsar magnetosphere. In general, values of f_Ω greater than 1 are consistent with the “fan-beam” emission from outer-magnetospheric gaps, whereas PC models tend to produce narrow beams, i.e., f_Ω ≪ 1.

The geometrical constraints from our radio-polarization measurements can be combined with information derived from the observed γ-ray light curve, to place more stringent limits on the allowed values of α and ζ. More specifically, we can use the measured P1–P2 peak separation of Δ ≈ 0.35 and the gap-thickness value for PSR J2043 + 2740, w = (L_sd/10³³ erg s⁻¹)^−1/2 ∼ 0.1, defined in Watters et al. (2009), to explore the allowed geometries that are consistent with the TPC and the OG model, based on the ATLAS maps of Watters et al. (2009). Figure 4 shows the α and ζ values allowed by the RVM fit of the P.A. in gray-scale contours. The green and pink points show the allowed geometries from the ATLAS of Watters et al. (2009), for a two-peaked gamma-ray profile and for a profile with a phase separation of 0.35–0.4 between the major peaks, respectively. The regions where all contours (from the radio data and from the γ-ray models) overlap are delineated with solid, black lines.

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For the TPC model, the overlap between the contours covers roughly the ranges α∼ 52°–57° and ζ ∼ 61°–68°. For the OG model, there are two separate regions of overlap between the radio contours and the ATLAS maps: these are roughly α∼ 62°–73° and ζ ∼ 74°–81°, and α∼ 72°–83° and ζ ∼ 60°–75°. The presence of overlapping regions in both TPC and OG contour plots does not allow us to exclude either of the two models. However, it is worth noting that the two outer peaks expected in the TPC geometry both have progressive leading edges and sharp trailing edges, whereas the OG model predicts symmetric peaks (see the Appendix of Watters et al. 2009). Therefore, the gamma-ray light curve shape we observe is consistent with the OG model prediction and inconsistent with the TPC.

Furthermore, as was mentioned earlier, the P1/P2 ratio decreases with energy. There are already several investigations of the energy dependence of P1 and P2 for Fermi pulsars (Abdo et al. 2010d, 2010e, 2010f). In all those cases, a double-peaked profile is observed that reveals a harder spectral index for P2 compared to that of P1. This indicates, for curvature radiation-dominated models, a higher accelerating electric field and/or a smaller radius of field-line curvature in the P2 emission region; future modeling can use these spectral variations to probe the underlying physics. Interestingly, the highest-energy γ ray is found nearer to P1 than to P2, which means that it is either part of the surviving photons from the respective cutoff process operating near the P1 region, or simply a background photon not associated with the pulsar.

4.1.1. Altitude of Emission

As was mentioned in the introduction, the high characteristic age of PSR J2043 + 2740 gives us an opportunity to investigate the claims for a correlation between γ-ray conversion efficiency and pulsar age, over a wider range of characteristic ages. However, it should be cautioned that any conclusions from such a study should take into account the dependency of both efficiency and characteristic age on the spin parameters, i.e., and . Therefore, other pulsar parameters that depend on the period and/or its time derivatives, for example, the surface polar field B or the field strength at the light cylinder radius, may well also exhibit a degree of correlation with the efficiency.

The γ-ray luminosity of a pulsar can be written as L_γ = 4πf_ΩG_>0.1 GeVD². Based on the measured γ-ray luminosities of non-recycled pulsars recently detected with Fermi, Abdo et al. (2010a) calculated the corresponding γ-ray efficiencies under the assumption of f_Ω = 1 and using distance estimates based either on the pulsar DM or known kinematic properties and SNR associations. A scatter plot of the γ-ray efficiency versus the characteristic age of non-recycled γ-ray pulsars with available distances is shown in Figure 5. In addition, the plot includes an upper limit on η, denoted with a solid triangle, corresponding to the efficiency of PSR J1836 + 5925. It should be noted that the efficiency of PSR J1836 + 5925 was recently estimated by Abdo et al. (2010d), assuming . However, we did not use their estimate in this work, because such an assumption introduces an artificial correlation between the pulsar's spin parameters and the γ-ray efficiency.

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Despite the large error bars on L_γ, Figure 5 reveals a correlation between η and τ_c, with older pulsars having on average higher γ-ray efficiencies compared to younger ones. The Spearman's rank coefficient for our data set is r_s = 0.57^+0.06_−0.07, which implies a positive correlation between efficiency and age. The probability of chance correlation given that value of r_s is only 0.11^+0.46_−0.09%.

In the plot of Figure 5, the data point corresponding to the highest characteristic age—without being an upper limit—is that of PSR J2043 + 2740 and is denoted by a solid black square. It is notable that the efficiency of PSR J2043 + 2740 is more comparable with that of pulsars that are at least 10 times younger, while the next three youngest pulsars after PSR J2043 + 2740, being roughly half as young, have efficiencies that are two to six times higher. For a number of pulsars in Figure 5, we have two distance estimates: from the pulsar DM and the NE2001 model, and from and independent method; the latter is based either on the Doppler shift of the H i lines of objects associated with the pulsar, combined with a rotation model for the Galaxy, or on trigonometric parallax, or on estimates by other means (Abdo et al. 2010a). For those pulsars, we have assigned two markers connected with a dashed line, each corresponding to the distance estimate from either method. For PSR J2043 + 2740, we can suppose that its association with the Cygnus Loop is true: this in turn would imply that its distance is ≈540 pc and, moreover, that its true age is roughly 12 kyr instead of 1.2 Myr. Therefore, if the association is true, then the marker for PSR J2043 + 2740 should be shifted to the position of the solid gray square in Figure 5, which would make the efficiency of this pulsar more consistent with the observed overall trend shown by the rest of the data points.

There are indeed a handful of examples for which a pulsar–remnant association has led to a significant revision to the pulsar's age: the case of PSR J0538 + 2817 and S147 has already been mentioned in the introduction; other examples include the 65 ms X-ray pulsar PSR J1811 − 1925, whose association with the Galactic SNR G11.2 − 0.3 suggested an age that was a factor 12 smaller than the spin-down age (Kaspi et al. 2001), and PSR B1951 + 32—one of the original six γ-ray pulsars detected with EGRET—with a characteristic age that is more than 1.5 times higher than the age of its birth site, the SNR CTB 80 (Migliazzo et al. 2002). The reason for such discrepancies between τ_c and the true age of a non-recycled pulsar is that the calculation of the characteristic spin-down age assumes in all cases that the pulsar was born with an initial period, P₀, that is negligible compared to that observed today. Evidently, in the above-mentioned cases, such an assumption does not hold, and those pulsars must have been born with a P₀ very close to the period observed today. Hence, in the case where P₀ ≪ P does not hold, the true age of the pulsar should be calculated from , where is the pulsar braking index. If P₀ ≪ P, τ can be approximated with τ_c. Under the usual assumption of n = 3, for pure-dipole magnetic braking, one can calculate τ for a range of P₀ values. By simply equating the true age of PSR J2043 + 2740 with the upper limit on the age of the Cygnus Loop, one finds that the only possibility of association with the remnant is if the pulsar was born with P₀ ≈ 95.6 ms.

A conclusive remark based on the above is that there is certainly some support for the pulsar–remnant association on the basis of γ-ray energetics. Nevertheless, it should be recognized that, as of yet, there exists no firm evidence for such an association and that a chance alignment between PSR J2043 + 2740 and the Cygnus Loop is as likely. Furthermore, it is important to note that a possible rejection of the above association does not provide indisputable support for the alternative distance to PSR J2043 + 2740, i.e., the one derived from the NE2001 model: it is quite possible, given the inaccuracies often associated with distances based on DM (e.g., Deller et al. 2009), that this pulsar is at an even greater distance than predicted by its DM. In fact, PSR J2022 + 2854, which is ≈45 away from PSR J2043 + 2740 in the sky, is at a DM distance of 2.7 kpc and has a comparable DM and RM to PSR J2043 + 2740 (RM_J2022+2854 = −75 rad m⁻²); while other nearby pulsars, e.g., J2113 + 2754 at 2 kpc, have a much lower RM (RM_J2113+2754 = −37 rad m⁻²).

The Cygnus Loop region has been searched by a number of authors for the central compact object, but so far there has been no conclusive evidence that led to a positive association (Thorsett et al. 1994; Ray et al. 1996; Miyata et al. 1998, 2001).

Figure 6 shows a radio map of the Cygnus Loop remnant relative to the position of PSR J2043 + 2740; the map was compiled from data from the Effelsberg 1.4 GHz Medium Latitude Survey (EMLS; Uyanıker et al. 1998, 1999; Reich et al. 2004). It has been proposed by Uyanıker et al. (2002), and was later supported with polarization observations at 6 cm (Sun et al. 2006), that the Cygnus Loop consists of two SNRs: the northernmost G74.3 − 8.4, centered at (α, δ) = (20^h5136, +31°3′), and G72.9 − 9.0, a blowout region in the southwestern rim of the Cygnus Loop, centered at (α, δ) = (20^h4956, +29°33′).

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Moreover, Miyata et al. (1998) have reported the discovery of a compact X-ray source, AX J2049.6 + 2939, near the center of G72.9 − 9.0 (see Figure 6). Although it may seem likely that AX J2049.6 + 2939 could be the central compact source associated with G72.9 − 9.0, follow-up observations with the ASCA and RXTE observatories showed significant X-ray variability, which makes the identification of AX J2049.6 + 2939 as an ordinary, rotation-powered pulsar unlikely (Miyata et al. 2001).

Given the lack of alternative candidates, the proximity of PSR J2043 + 2740 (≈15 outside the edge of G72.9 − 9.0) makes this pulsar the only possible prospect for an association. Ultimately, a conclusion to whether PSR J2043 + 2740 is associated with the Cygnus Loop can be drawn by measuring the pulsar's proper motion. One method of estimating pulsar proper motions is through pulsar timing. We examined 7 years of timing observations with the Lovell telescope, at 1.4 GHz. We fitted 234 TOAs for the pulsar period and its first and second time derivatives. The post-fit timing residuals displayed the typical, long-term (i.e., “red”), correlated timing signature often seen in the TOAs of non-recycled pulsars: this phenomenon is commonly referred to as “timing noise” (see Hobbs et al. 2004, 2010). Younger and more energetic pulsars often exhibit a higher amount of timing noise than older ones. PSR J2043 + 2740 is ranked amongst the pulsars with the highest amount of timing noise, a fact which is demonstrated by this pulsar's high and significant value of s⁻³ (compare with Table 1 of Hobbs et al. 2010).

The fitting procedure that can be used to derive the pulsar proper motion from timing data only provides correct results when applied to statistically “white” data. Therefore, some form of pre-whitening needs to be applied to effectively remove the timing noise mentioned above (e.g., Hobbs et al. 2006). We attempted the standard technique of fitting harmonically related sinusoids, but given the strength of the timing noise in this pulsar we were unable to pre-whiten the timing residuals sufficiently to obtain a plausible fit for proper motion.

Another method for inferring the pulsar's proper-motion direction is through our radio-polarization measurements (Section 2.2). It has been claimed that there is strong evidence for a correlation between the projected spin-axis orientations of pulsars and their respective velocity-vector directions (Johnston et al. 2005, 2007). It has also been suggested that such a correlation—if there is indeed a physical mechanism that aligns pulsar spin axes with their velocities—should vanish for old pulsars, as the Galactic gravitational potential will have had enough time to alter the velocities of those pulsars. On the other hand, the young population of pulsars should display a stronger case for alignment.

The position angle P.A.₀ defines the orientation—relative to our polarization dipoles—of the plane of linear polarization of the pulsar's emission, at the closest approach of the magnetic pole to our line of sight. Assuming that the plane of polarization coincides with the plane defined by the magnetic field lines and the magnetic axis, P.A.₀ should also correspond to the orientation of the spin axis, projected on our field of view, i.e., P.A._r ≡ P.A.₀. Note that the calculation of the P.A. from Stokes Q and U can only provide a “headless” vector with 180° ambiguity. In addition, as was shown by Backer et al. (1975) and Manchester et al. (1975), pulsar emission can occur in two orthogonal modes, i.e., the polarization plane can either coincide with the field-line–magnetic-axis plane or be perpendicular to it. Therefore, there is an additional 90° ambiguity in the projected direction of the spin axis.

If we assume that the scenario of alignment between the spin axis and velocity is true for young pulsars and that PSR J2043 + 2740 was indeed born within the bounds of the Cygnus Loop—making it a young pulsar—then we expect that the spin-axis orientation is, within the measurement errors, directed toward the SNR. Our polarization measurements show that the most favorable case to the above scenario, given the 90° ambiguity, gives a position angle for the spin axis of 17° north through east. The spin-axis orientation relative to the Cygnus Loop is shown in Figure 6, drawn at the pulsar's position. From the figure, it is immediately evident that the spin-axis orientation is significantly offset from the direction to the center of G72.9 − 9.0, the offset being ≈19°.

A direct implication of the above offset could be that our assumptions are not valid and that PSR J2043 + 2740 is not associated with the Cygnus Loop. Another possibility is that PSR J2043 + 2740 was born in the supernova but the velocity vector is not aligned with the pulsar's spin axis: it is true that in the work of Johnston et al. (2005) the offsets between the velocity vectors and the spin axes show a significant spread. However, from the same work, Johnston et al. (2005) derive that roughly 90% of the pulsars have an offset of <19° between the spin-axis orientation and the velocity vector, which makes the measured offset for PSR J2043 + 2740 an unlikely product of any underlying alignment mechanism leading to the observed distribution of offsets.

In conclusion, the above arguments for the pulsar–remnant association, based on the pulsar orientation and its inferred proper-motion direction, do not appear to favor the Cygnus Loop as the birthplace of PSR J2043 + 2740. However, all of these arguments are heavily based on uncertain assumptions; a proper-motion measurement is still required for an unequivocal verdict.

Recently, it was proposed that a large part of the observed timing noise of PSR J2043 + 2740, as well as that of other pulsars, is due to pulse-shape variations (Lyne et al. 2010); those variations appear to be correlated with changes in . Moreover, it was suggested that one can fit for such pulse-shape variations and mitigate the large variations in the timing residuals by a significant factor. In particular, PSR J2043 + 2740 shows major pulse variations that cause 100% change in the pulse's FWHM on time scales of ≈200 days. Our data set, covering nearly 500 days, is certainly affected by those changes; still, for a proper-motion fit that would use years of data—as is needed to obtain a precise measurement—such an effect dominates over the proper-motion signature, i.e., for PSR J2043 + 2740, the rms of the timing residuals over 4000 days is >1 s (e.g., see Figure 1 of Lyne et al. 2010). In conclusion, such a method is certain to assist proper-motion estimates through pulsar timing for a large number of “noisy” pulsars, such as PSR J2043 + 2740.

Alternatively, a direct measurement of the pulsar's proper motion can be made with VLBI observations. If the pulsar was born 12 kyr ago in the supernova explosion that created G72.9 − 9.0, then the angular separation between the center of the SNR and the pulsar, i.e., ≈23, translates to a transverse velocity of V_⊥ ∼ 1770 km s⁻¹, assuming both pulsar and remnant are at 540 pc distance. Indeed, this corresponds to a large value of proper motion (≈690 mas yr⁻¹), with the highest inferred transverse velocities published so far being V_⊥ ∼ 1600 km s⁻¹ (Hobbs et al. 2005). We have performed TEMPO2 simulations that generate fake TOAs having a proper-motion signature corresponding to a pulsar at the position of PSR J2043 + 2740, with 690 mas yr⁻¹. Fitting those TOAs with the pulsar's spin parameters but setting the proper motion to zero reveals the characteristic periodic sine wave, modulated with time difference from the reference position epoch. The maximum amplitude of the simulated wave over a few thousand days, due to the proper motion, was ∼10 ms (see also Figure 8.2(d) in Lorimer & Kramer 2005). This otherwise large amount of residual rms is still swamped, for the case of PSR J2043 + 2740, by the much larger timing noise of this pulsar. However, if PSR J2043 + 2740 possesses such a large proper motion, it will be easily measurable with VLBI, which will conclusively confirm or reject the pulsar–remnant association.