DISCOVERY OF A FAINT X-RAY COUNTERPART AND A PARSEC-LONG X-RAY TAIL FOR THE MIDDLE-AGED, γ-RAY-ONLY PULSAR PSR J0357+3205

Our Chandra observation of PSR J0357+3205 was split between two consecutive satellite revolutions. The first observation started on 2009 October 25 at 00:56 UT and lasted 29.5 ks; the second observation started on 2009 October 26 and lasted 47.1 ks. The two observations are almost co-aligned, with very similar pointing directions and satellite roll angles. The target position was placed on the back-illuminated ACIS S3 chip. The time resolution of the observation was 3.2 s. The VFAINT exposure mode was adopted. We retrieved "level 1" data from the Chandra X-ray Center Science Archive and we generated "level 2" event files using the Chandra Interactive Analysis of Observations (CIAO v.4.2) software.¹⁰ We also produced a combined event file using the merge_all script.¹¹

Deep optical and near-infrared images of the field of PSR J0357+3205 were collected with the 4 m Mayall Telescope as a part of our joint Chandra–NOAO program. Optical observations in the V band ("V Harris" filter, λ = 5375 Å, Δλ = 945.2 Å) were performed using the large-field (36' × 36') Mosaic CCD Imager (Jacoby et al. 1998) on 2009 November 10. The sky was mostly clear, with a few thin cirri. Seeing was always better than 11. We obtained a first set of five exposures of 10 minutes each and a second set of 18 exposures of 12 minutes each, for a total integration time of 4 hr 26 minutes. Fifty-five percent of the observations were performed in dark conditions, and 45% had a partially (∼43%) illuminated Moon, about 85° away from the target position. We used a standard five-point dithering pattern. We performed standard data reduction (bias subtraction and flat fielding), CCD mosaic, and image co-addition using the package mscred available in IRAF.¹²

In our resulting co-added image, point sources have an FWHM of ∼10 close to the expected target position. An astrometric solution was derived using more than 1000 stars from the Guide Star Catalog 2 (GSC2.3; Lasker et al. 2008) with an rms deviation of ∼025 across the whole field of view. Following Lattanzi et al. (1997), and taking into account the mean positional error in the GSC2.3 source coordinates as well as the uncertainty on the alignment of GSC2.3 with respect to the International Celestial Reference Frame (Lasker et al. 2008), our absolute astrometric accuracy is 029. In view of the non-optimal sky conditions, photometric calibration of the image was performed using a set of more than 400 unsaturated sources, also listed in the GSC2.3, taking into account the transformation from the photographic band to the Johnson band (Russell et al. 1990) assuming a flat spectrum as a function of frequency. The rms of the fit is ∼0.12 mag.

Near-infrared observations were performed at Mayall on 2010 February 2, using FLAMINGOS (Elston et al. 2003), having a field of view of 10' × 10', using the Ks filter (λ = 2.16 μm, Δλ = 0.31 μm). Sky conditions were not optimal, with passing thin to moderate cirrus clouds. Seeing was good, always better than 09. To allow for the subtraction of the variable IR sky background, observations were split in 15 sequences (stacks) of short dithered exposures with an integration time of 30 s. Data reduction was performed using MIDAS¹³ and SciSoft/ECLIPSE¹⁴ packages. The near-IR raw science images have been linearized, dark subtracted, and flat fielded. The flat fields were generated from science frames via median stacking to ensure that the flat field was stable over the 2 hr observing time. Data consist of 15 stacks, each one composed of 16 or 25 jittered images on a 4 × 4 or 5 × 5 grid. In the first step, the reduced raw frames of a stack were co-added using the SciSoft jitter command. The sky was subtracted as a moving average from the frames before the co-addition step. A bad pixel map, derived from the flat field, was used for masking these pixels. For some of the stacks, the jitter offset values required manual adjustments in order to improve the image alignment. As the ambient conditions became less stable in the second half of the observing run, a brighter correlation star had to be used further away from the center of the field of view. As a last step, the 15 intermediate products were co-added to generate the final deep image. This image is composed of 254 raw frames and corresponds to a total integration time of 2 hr 7 minutes. An astrometric solution was computed (∼02 accuracy) based on a set of stars from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) catalog. Photometric calibration, owing to poor sky conditions, was performed on the image using a set of 35 stars also listed in the 2MASS catalog, with an rms of 0.12 mag.

Additional optical observations of PSR J0357+3205 in the B (λ = 4298 Å, Δλ = 1065 Å), R (λ = 6380 Å, Δλ = 1520 Å), and I (λ = 8063 Å, Δλ = 1500 Å) bands were obtained in dark time with the Wide Field Camera (WFC) on the 2.5 m INT on the nights of 2010 January 16–17, with seeing in the 11–13 range (see A. Shearer et al. 2011, in preparation), for a total integration time of 6000 s in each band. The WFC is a mosaic of four thinned 2048 × 2048 pixel CCDs, with a pixel size of 033 and a full field of view of 342 × 342. To compensate for the 1' gaps between the CCDs and for the fringing in the I band, observations were split in a sequence of 600 s exposures with a five-point dithering pattern. Data reduction was also performed with IRAF. Our astrometric solution was computed using 13 USNOB stars¹⁵ with 026 accuracy. For photometric calibration, 45 USNO-B1 stars (Monet et al. 2003) were used for I-band images, 30 for B, and 16 for R.

In order to identify the X-ray counterpart of the γ-ray pulsar, we searched the most recent Fermi-LAT timing error circle for X-ray sources showing the expected signature of isolated neutron stars, i.e., a very high X-ray-to-optical flux ratio.

We generated an X-ray image in the 0.5–6 keV energy range using the ACIS original pixel size (0492). We ran a source detection using the wavdetect¹⁶ task, with wavelet scales ranging from 1 to 16 pixels, spaced by a factor of $\sqrt{2}$. A detection threshold of 10⁻⁵ was selected so as not to miss faint sources. The Fermi-LAT timing error circle for PSR J0357+3205 is centered at R.A. = 03:57:52.5, decl. = +32:05:25 and has a radius of 18'' (Ray et al. 2010). Only one X-ray source, positioned at R.A.(J2000) = 03:57:52.32, decl. = +32:05:20.6, was detected within such a region, with a background-subtracted count rate of (6.3 ± 0.3) × 10⁻³ counts s⁻¹ in the 0.5–6 keV energy range (see Figure 1). In order to check the accuracy of the Chandra/ACIS absolute astrometry, we cross-correlated positions of ACIS sources detected at >4.5σ within 3 arcmin from the aim point with astrometric catalogs. We found two coincidences in the GSC2.3, with offsets of 015–03. One of those two sources is also listed in 2MASS, with a 015 offset with respect to the Chandra position. Although we could not derive an improved astrometric solution, such an exercise suggests that the Chandra astrometry is not affected by any systematics in our observations. Thus, we attached to the coordinates of our candidate counterpart a nominal positional error of 025 (at a 68% confidence level¹⁷). No coincident optical/infrared sources were found in our deep images collected at Kitt Peak, down to 5σ upper limits V >26.7, Ks >19.9 (the inner portion of the field, as seen in the V band, is shown in Figure 2). The INT observation allows us to set 5σ upper limits of B >25.86, R >25.75, and I >23.80 (see Figure 3). Assuming the best-fit spectral model for the X-ray source (see below), the corresponding X-ray-to-optical (V band) flux ratio is F_X/F_V > 520, while the X-ray-to-near-infrared (Ks band) flux ratio is F_X/F_Ks > 30. Thus, positional coincidence coupled to very high X-ray-to-optical flux ratio prompts us to conclude that our X-ray source is the counterpart of PSR J0357+3205.

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To evaluate the source spectrum, we extracted photons within a 15 arcsec radius (561 counts in the 0.2–6 keV range, with a background contribution <0.4%) and we generated an ad hoc response matrix and effective area file using the CIAO script psextract.¹⁸ We used the C-statistic approach (see, e.g., Humphrey et al. 2009) implemented in Xspec¹⁹ (requiring neither spectral grouping nor background subtraction), which is well suited to study sources with low photon statistics. Errors are at a 90% confidence level for a single parameter. The pulsar emission is well described (the p-value, i.e., probability of obtaining the data if the model is correct, is 0.62) by a simple power-law model, with a steep photon index (Γ = 2.53 ± 0.25), absorbed by a hydrogen column density N_H = (8 ± 4) × 10²⁰ cm⁻². A blackbody model yields a poor fit (p-value <0.00005). Assuming the best-fit power-law model, the 0.5–10 keV observed flux is (3.9^+0.7 _{− 0.6}) × 10⁻¹⁴ erg cm⁻² s⁻¹. The unabsorbed 0.5–10 keV flux is 4.7 × 10⁻¹⁴ erg cm⁻² s⁻¹.

The limited statistics prevented us from constraining a more complex composite, thermal-plus-non-thermal model, due to spectral parameter degeneracy (e.g., N_H versus the normalization of the pulsar emission components). To ease the problem, we set an independent upper limit to the N_H. The total Galactic absorption in the direction of the target was (7–10) × 10²⁰ cm⁻² (Dickey & Lockman 1990; Kalberla et al. 2005). Since such values, based on H i surveys, could differ significantly with respect to the actual X-ray absorption, we used our X-ray data to get an independent N_H estimate. Our brightest point source (source "A," see Figure 4) is a bona fide active galactic nucleus (AGN), with a power-law spectrum (Γ = 1.75 ± 0.15), a flux of ∼1.1 × 10⁻¹³ erg cm⁻² s⁻¹, and a F_X/F_opt ratio of ∼11. Its absorbing column is N_H = (1.0 ± 0.3) × 10²¹ cm⁻². Thus, we can assume conservatively N_H = 1.3 × 10²¹ cm⁻² as the maximum possible value for the absorption toward the target. Such a constraint on N_H allowed us to estimate upper limit temperatures for any thermal emission from PSR J0357+3205 originating (1) from a hot polar cap and (2) from the whole neutron star surface. Assuming standard blackbody emission and the standard polar cap radius (r_PC = (2πR³/cP)^1/2 = 320 m), we obtain kT <122 eV (at 3σ confidence level) for a 500 pc distance. Similarly, for an NS radius of 13 km, we obtain kT < 35 eV as a limit to the temperature of the whole surface of the star (at 3σ confidence level). Blackbody radii and temperature reported above are the values as seen by a distant observer.

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Our Chandra data unveil the existence of a peculiar X-ray feature in the field of PSR J0357+3205. An extended structure of diffuse X-ray emission, apparently protruding from the pulsar position, is seen in the ACIS image, extending >9 arcmin in length and ∼1.5 arcmin in width (see Figure 4). A total of 1550 ± 75 background-subtracted counts in the 0.5–6 keV band were collected from the feature (the "tail," hereafter).

We studied the morphology of the tail, extracting surface brightness profiles on different regions (see Figure 5). First, we searched for diffuse emission in the pulsar surroundings, by comparing the source intensity profile to the expected ACIS point-spread function (PSF). Assuming the pulsar best-fit spectral model, we simulated a PSF using the ChaRT²⁰ and MARX²¹ packages. Results in the 0.5–6 keV energy range are shown in Figure 6, where the lack of any significant diffuse emission within 20'' of the pulsar position is apparent. Then, we extracted exposure-corrected, background-subtracted surface brightness profiles on a larger angular scale. Along the main (northwest to southeast) axis, the tail emerges from the background ∼20 arcsec away from PSR J0357+3205, shows a broad maximum after ∼4', and then fades away at more than 9' from the pulsar (see Figure 7). A possible local minimum in the surface brightness is also seen at ∼2' from the pulsar position. In the direction orthogonal to the main axis, the profile shows a sharper edge toward northeast (rising to the maximum within 15'') and a shallower decay to the southwest (fading to the background in ∼70''), as shown in Figure 8. We also extracted energy-resolved images in "soft" (0.5–1.5 keV) and "hard" (1.5–6 keV) energy bands. However, no significant differences in the brightness profiles were observed (see Figures 7 and 8).

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Spectral analysis of the tail emission was hampered by the low signal-to-noise ratio. In the extraction region, background accounts for ∼57% of the total counts in 0.5–6 keV. A background spectrum was extracted from a source-free region northeast of the tail. Response and effective area files were generated using the CIAO specextract script.²² The spectrum of the tail is described well (χ²_ν = 1.0, 70 degrees of freedom) by a non-thermal emission model (power-law photon index Γ = 1.8 ± 0.2), absorbed by a column N_H = (2.0 ± 0.7) × 10²¹ cm⁻². Confidence contours for N_H and photon index of the diffuse feature, compared to the ones of the pulsar, are shown in Figure 9. Fixing N_H to 8 × 10²⁰ cm⁻² (the best-fit value for the pulsar counterpart) yields a statistically acceptable fit (χ²_ν = 1.1, 71 degrees of freedom), with a photon index Γ = 1.55 ± 0.15. Adopting the latter model, the total observed flux of the tail in the 0.5–10 keV energy range is (2.4 ± 0.4) × 10⁻¹³ erg cm⁻² s⁻¹, corresponding to an average surface brightness of 2.5 × 10⁻¹⁴ erg cm⁻² s⁻¹ arcmin⁻². The unabsorbed flux in the same energy range is 2.9 × 10⁻¹³ erg cm⁻² s⁻¹. We note that a thermal bremsstrahlung model also fits the data well (χ²_ν = 1.0, 70 degrees of freedom) with an absorbing column N_H = (1.4 ± 0.7) × 10²¹ cm⁻², but requires an unrealistic temperature, kT = 5.4 ± 1.7 keV.

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Spatially resolved spectroscopy was also performed, using two separate extraction regions, both along the tail and across it, assuming a power-law model. However, no significant spectral differences were found, which is consistent with the results of our energy-resolved imaging reported above. For instance, dividing the tail in two sections, for N_H = 8 × 10²⁰ cm⁻², we found photon indexes Γ = 1.45 ± 0.15 in the first half of the tail and Γ = 1.60 ± 0.15 in the second half.

No point sources are detected superimposed to the tail, with the exception of two objects located close to the southeast end (sources "1" and "2" in Figure 4). X-ray spectroscopy points to non-thermal emission spectra for such sources. Both sources have very likely optical counterparts in our ground-based images. The resulting X-ray-to-optical flux ratio is F_X/F_V ∼ 13 and F_X/F_V ∼ 4.5 for sources 1 and 2, respectively. Such results allow us to conclude that they are unrelated extragalactic sources. Our ground-based images do not show any bright optical source possibly associated with the tail, or hints of correlated, diffuse emission. We also retrieved and analyzed public data at radio wavelengths from the NRAO VLA Sky Survey (NVSS; Condon et al. 1998). The images at 1.4 GHz do not show any counterpart for the tail. We could set upper limits of 6.1 mJy to the tail radio emission over the whole extension of the X-ray feature (T. Cheung 2010, private communication). The tail has also been detected by Suzaku (Y. Kanai 2010, private communication), in a 40 ks long observation, although such data could not resolve its shape, or yield a better characterization of its spectrum.

Our multiwavelength campaign allowed us to identify the faint X-ray counterpart of the γ-ray-only pulsar PSR J0357+3205. Bright in γ-rays (Abdo et al. 2009a, 2009b), with a γ-ray-to-X-ray flux ratio of F_γ/F_X ∼ 1300, PSR J0357+3205 is an unremarkable X-ray source. Although the small photon statistics do not allow us to draw firm conclusions, the non-negligible interstellar absorption points to a distance of a few hundred parsecs for the source, in broad agreement with the value of ∼500 pc estimated by scaling its γ-ray flux, using the γ-ray pseudo-luminosity relation by Saz Parkinson et al. (2010).

The ACIS spectrum is consistent with a purely non-thermal origin of the X-ray emission. The 0.5–10 keV luminosity (at 500 pc) is L_X = 1.4 × 10³⁰ erg s⁻¹, accounting for ∼2.4 × 10⁻⁴ of the pulsar rotational energy loss $\dot{E}_{{\rm rot}}$, in broad agreement with the dependence of the X-ray luminosity of rotation-powered pulsars on the spin-down luminosity (Becker & Trümper 1997; Possenti et al. 2002; Kargaltsev & Pavlov 2008a). The photon index is significantly steeper than the typical value of ∼1.8 observed for middle-aged pulsars (De Luca et al. 2005).

No thermal emission from the neutron star surface was detected. The 3σ upper limit to the bolometric luminosity is ∼5 × 10³¹ erg s⁻¹. Such a limit can be compared to the bolometric luminosity of the well-studied surface thermal emission of the Three Musketeers, which have a characteristic age similar to that of our target. The upper limit to the thermal emission from PSR J0357+3205 is a factor of 10 lower than the bolometric luminosity of PSR B0656+14 and PSR B1055−52²³ (De Luca et al. 2005), but it is comparable to the luminosity of Geminga (Caraveo et al. 2004). Although PSR J0357+3205 turns out to be the coldest neutron star in its age range (0.1–1 Myr), the upper limit to its thermal emission is consistent with the expectations of several cooling models (see, e.g., Tsuruta et al. 2009; Page et al. 2009). On the other hand, the apparent lack of emission from the polar caps is also interesting, since PSR J0357+3205 is a bright γ-ray pulsar, channeling about 40% of its spin-down luminosity in γ-rays of magnetospheric origin; thus, polar cap reheating by "return currents" in the magnetosphere would be expected. Our limit to the temperature of a hot polar cap points to a bolometric luminosity L_PC < 5 × 10³⁰ erg s⁻¹, a factor of >5 lower than the polar cap luminosity for PSR B0656+14 and PSR B1055−52 (De Luca et al. 2005), but a factor of ∼10 larger than the polar cap luminosity of Geminga (Caraveo et al. 2004). The upper limit is a factor of a few lower than the luminosity expected by heating models based on return currents of e⁺/e⁻ generated above the polar caps by curvature radiation photons, but it is consistent with expectations for polar cap heating due to bombardment by particles created only by inverse Compton scattered photons (Harding & Muslimov 2001, 2002). PSR J0357+3205 is close to the death line for production of e⁺/e⁻ by curvature radiation photons (Harding & Muslimov 2002), which could explain the reduced polar cap heating. As a further possibility, the system's viewing geometry could play some role, as in the case of Geminga, where the emitting area and luminosity of the thermally emitting polar cap suggested an almost aligned rotator, seen at a high inclination angle (Caraveo et al. 2004; De Luca et al. 2005).

When compared to other well-known middle-aged rotation-powered pulsars, the X-ray spectrum of PSR J0357+3205 is remarkably different. Indeed, it is reminiscent of a number of older (τ_C ∼ 10⁶–10⁷ yr) pulsars, e.g., PSR B1929+10 (Becker et al. 2006), B1133+16 (Kargaltsev et al. 2006), B0943+10 (Zhang et al. 2005), B0628+28 (Becker et al. 2005). A non-thermal origin for the bulk of the X-ray emission from such pulsars was proposed by Becker et al. (2004, 2006), although such a picture was questioned, e.g., by Zavlin & Pavlov (2004) and Misanovic et al. (2008), who preferred a composite, thermal-plus-non-thermal spectral model.

The morphology of the tail, apparently protruding from PSR J0357+3205 and smoothly connected to the pulsar counterpart strongly argues for a physical association of the two systems. This is also supported by the lack of any other source possibly related to the extended feature. Sources "1" and "2" are extragalactic objects. An interpretation of the feature as an AGN jet, associated with, e.g., Source 2, can be safely discarded, owing to the lack of radio emission for both the point source and the diffuse feature, at variance with all known AGN jets (Harris & Krawczynski 2006). Furthermore, the angular extent of the feature would imply an unrealistic physical size, unless the source is quite local (a huge 200 kpc long jet would imply an angular scale distance of order 80 Mpc, assuming standard cosmological parameters), which would call for a rich multiwavelength phenomenology (the host galaxy itself—with an angular scale well in excess of 1 arcmin—should be clearly resolved in our ground-based optical images).

Assuming an association of the feature to PSR J0357+3205, the observed extension of the tail, at a distance of 500 pc, would correspond to a physical length of ∼1.3 pc (assuming no inclination with respect to the plane of the sky).

A few elongated "tails" of X-ray emission associated to rotation-powered pulsars have been discovered by Gaensler et al. (2004), McGowan et al. (2006), Becker et al. (2006), and Kargaltsev & Pavlov (2008b). Such features are interpreted within the framework of bow-shock, ram-pressure-dominated, pulsar wind nebulae (PWNe; see Gaensler & Slane 2006 for a review). If the pulsar moves supersonically, shocked pulsar wind is expected to flow in an elongated region downstream of the termination shock (basically, the cavity in the interstellar medium (ISM) created by the moving neutron star and its wind), confined by ram pressure. X-ray emission is due to synchrotron emission from the wind particles accelerated at the termination shock, which is typically seen (if angular resolution permits) as the brightest portion of the extended structure (see, e.g., Kargaltsev & Pavlov 2008b), as expected from MHD simulations (Bucciantini 2002; van der Swaluw 2003; Bucciantini et al. 2005).

Although for our radio-quiet pulsar we have no information about the proper motion, the bow-shock PWN scenario would seem the most natural explanation. Of course, such a picture would suggest for PSR J0357+3205 a large space velocity aligned with the tail, in the direction opposite of the tail extension. If this is the case, the pulsar would be moving almost parallel to the plane of the Galaxy, which would suggest that it was born out of the Galactic plane, at a height of order 140d₅₀₀ pc (where d₅₀₀ is the distance to the pulsar in units of 500 pc), possibly from a "runaway" high mass star (Mason et al. 1998).

The luminosity of the tail in the 0.5–10 keV energy range (assuming d = 500 pc) is 8.8 × 10³⁰ erg s⁻¹, corresponding to a fraction 1.5 × 10⁻³ of the pulsar spin-down luminosity. Indeed, such a value is fully compatible with that measured for other pulsars, which channel into their tails 10⁻² to 10⁻⁴ of their rotational energy loss. Synchrotron cooling of the particles injected at the termination shock induces a significant softening of the emission spectrum as a function of the distance from the pulsar in bow-shock PWNe. For the tail of PSR J0357+3205, we do not have firm evidence for such a spectral variation.

However, explaining the tail of PSR J0357+3205 within the bow-shock PWN frame is not straightforward. A first difficulty arises from energetic requirements for the emitting particles—indeed, the hypothesis that the observed X-rays from the tail are due to synchrotron emission is somewhat challenging for a pulsar with such a low $\dot{E}_{{\rm rot}}$. As in the case of PSR B1929+10, discussed by Becker et al. (2006) and de Jager & Djannati-Ataï (2009), the problem lies with the maximum energy of the particles injected into the PWN. Particle acceleration mechanisms in PWNe are not yet fully understood. The maximum energy to which electrons can be accelerated (via acceleration of the pulsar wind and then re-acceleration at the termination shock) is expected to be a fraction of the polar cap potential (∼0.1 for the Crab, see de Jager et al. 1996; see also de Jager & Djannati-Ataï 2009, Bandiera 2008). According to Goldreich & Julian (1969), the maximum potential drop between the pole and the light cylinder (in an aligned pulsar) is $\Delta \Phi = (3\dot{E}_{{\rm rot}}/2c)^{1/2}$. For PSR J0357+3205, this would correspond to electron acceleration in the pulsar magnetosphere up to a maximum Lorentz factor γ_max ∼ 10⁸, which can be considered to be an upper limit for the electrons injected in the PWN. The characteristic energy of synchrotron photons is ∼0.5B₋₅γ²₈ keV, where B₋₅ is the ambient magnetic field in units of 10 μG and γ₈ is the Lorentz factor of the radiating electrons in units of 10⁸. It is clear that, in order to produce bright emission at few keV, the typical Lorentz factor of the electrons in the tail has to be of the same order of γ_max, implying the presence of e⁺/e⁻ accelerated at the highest possible energy, as well as an ambient magnetic field as high as ∼50 μG. If this is the case, it is possible to estimate the synchrotron cooling time of the emitting electrons as $\tau _{{\rm sync}}\,\sim 100\,(B/50 \, \rm \mu G)^{-3/2}\,(E/1\,\rm keV)^{-1/2}$ yr. Coupling this value with the estimated physical length of the feature yields an estimate of the bulk flow speed of the emitting particles of ∼15, 000 km s⁻¹, assuming no inclination with respect to the plane of the sky. Such a value is consistent with results for other bow-shock PWNe (Kargaltsev & Pavlov 2008b).

A second difficulty for the bow-shock interpretation arises owing to the lack of diffuse emission surrounding the pulsar. Bright emission from the wind termination shock should be seen there as the maximum surface brightness portion of the diffuse feature, as observed in all other known cases (see, e.g., Gaensler et al. 2004; McGowan et al. 2006; Kargaltsev & Pavlov 2008b). As a possible way out, we evaluate under what conditions the termination shock could be unresolved by Chandra. Assuming standard relations (Gaensler & Slane 2006), the distance between the pulsar and the head of the termination shock is expected to be $r_S=(\dot{E}_{{\rm rot}}/4 \pi c \rho _{\rm ISM} v^2_{\rm PSR})^{1/2}$, where ρ_ISM is the ambient density and v_PSR is the pulsar space velocity. For PSR J0357+3205, r_s ∼ 10¹⁶v⁻¹_{PSR, 100}n^−1/2_{ISM, 1} cm, where v_{PSR, 100} is the pulsar space velocity in units of 100 km s⁻¹ and n_{ISM, 1} is the ambient number density in units of 1 cm⁻³. At a distance of 500 pc, this corresponds to ∼13v⁻¹_{PSR, 100}n_{ISM, 1}^−1/2. The surface of the termination shock (in the hypothesis of an isotropic pulsar wind) should assume an elongated shape, extending ∼6r_s (∼8''v⁻¹_{PSR, 100}n_{ISM, 1}^−1/2 at 500 pc) behind the pulsar. The termination shock could hide within the PSF of the pulsar if 6r_s < 05, which would require an unrealistically large ambient number density (of order several hundreds per cm³), and/or a pulsar space velocity of at least 1000 km s⁻¹. Anisotropies in the pulsar wind could also play some role. Such a picture would suggest that a significant fraction of the flux of the point source is due to emission from the wind termination shock.

A further problem with the bow-shock picture is related to the brightness profile of the tail, which is remarkably different from what is observed for all other diffuse structures interpreted as ram-pressure dominated PWNe. Figures 3 and 4 clearly show that the surface brightness grows as a function of angular distance from PSR J0357+3205 and reaches a broad maximum ∼4' away from the pulsar, while all the elongated structures imaged so far have their peaks close to their parent pulsar position (although localized, bright "blobs" along the tails have been observed; see, e.g., Kargaltsev & Pavlov 2008b and interpreted as due to kink instabilities in the particle flow). Invoking geometric effects, such as bending of the tail along the line of sight, producing limb brightening (higher column density of emitting particles), would require ad hoc assumptions for the tail's three-dimensional structure. Since no plausible explanations for the origin of such bending are apparent, we discard such a possibility. The lack of any significant spectral evolution along the tail ultimately prevents us from drawing conclusions about the physical nature of its peculiar profile.

The "asymmetric" brightness profile of the tail in the direction perpendicular to its main axis (with its sharp northeastern edge and its shallower decay toward the southwest) is also remarkably different from what is observed for any other diffuse structure interpreted as a ram-pressure-dominated PWN, but is reminiscent of the case of the peculiar diffuse X-ray feature associated with PSR B2224+65, which powers the "guitar" nebula (Hui & Becker 2007; Johnson & Wang 2010). The extended feature seen there is remarkably misaligned (by 118°) with respect to the direction of the pulsar proper motion (Hui & Becker 2007) and therefore it has been interpreted in a different frame, either as a "magnetically confined" PWN (Bandiera 2008) or as a jet from the pulsar (Johnson & Wang 2010). Both pictures naturally predict the feature to be brighter in the leading edge (the one in the direction of the proper motion), where "fresh" electrons are injected. The profile in the trailing edge is expected to fade smoothly, dominated by cooling of the electrons deposited by the moving source (i.e., the feature is a "synchrotron wake" along its minor axis). Thus, if this is the case, the proper motion of the pulsar should not be aligned with the tail main axis and the tail itself should display a proper motion. In view of the lack of information about the pulsar proper motion, it is premature to discuss this scenario any further. We note, however, that both pictures are not free from difficulties. For instance, the jet explanation cannot easily explain the lack of any appreciable bending of the structure due to ram pressure from the ISM. On the other hand a magnetically confined PWN would require a very intense (50 μG), ordered ambient magnetic field (see Bandiera 2008; Johnson & Wang 2010, for further details on such pictures for the case of PSR B2224+65). Moreover, the broad maximum at a large distance from the pulsar would not be accounted for easily in these pictures (which, similarly to the bow-shock picture, predict a brightness peak close to the pulsar).