OPTICAL–ULTRAVIOLET SPECTRUM AND PROPER MOTION OF THE MIDDLE-AGED PULSAR B1055−52^*

The most recent published radio coordinates of PSR B1055−52 (Newton et al. 1981),

$$\begin{equation} \alpha =10^{\rm h} 57^{\rm m} 58\mbox{$.\!\!^{\mathrm s}$}84\pm 0\mbox{$.\!\!^{\mathrm s}$}03, \quad \delta = -52^\circ 26\mbox{$^\prime $}56\farcs 3 \pm 0\farcs 3, \end{equation} \tag{ 1 }$$

are for the epoch of 1978.13 (equinox J2000). These coordinates, without proper motion values, are reported in the ATNF radio pulsar catalog⁶ (Manchester et al. 2005), and they were also used by Weltevrede et al. (2010) to compute the PSR B1055−52 radio ephemeris for the timing of the Fermi γ-ray observations. Since pulsars are known to have high velocities, PSR B1055−52 could move a few arcseconds in the 30 year span between the epochs of the radio position and our HST observations (2008.12 and 2008.18). For instance, for an average radio pulsar velocity of 400 km s⁻¹ (e.g., Hobbs et al. 2005), we would expect a proper motion of 110 d⁻¹₇₅₀ mas yr⁻¹ and a position shift of 34 d⁻¹₇₅₀ in 30 years, in an unknown direction. Such an uncertainty hampers a direct identification based on the pulsar's position only. Fortunately, the presence of Star A in the immediate vicinity of the pulsar can be used for the pulsar identification.

The 10'' × 10'' cutout of the SBC image of the PSR B1055−52 field (left panel of Figure 1) shows the only two objects detected in the entire SBC FOV, separated by a distance of 441. From the comparison of this image with that obtained by M+97 with FOC, it is natural to assume that the northwestern and southeastern objects are Star A and the candidate pulsar counterpart, respectively.

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Indeed, the nominal SBC coordinates of the northwestern source are α = 10^h57^m58790 and δ = −52°26''5304. The most accurate coordinates of Star A, α_A = 10^h57^m58728 ± 0004, and δ_A = −52°26''5245 ± 006, at the epoch of our HST observation, can be derived from the third US Naval Observatory CCD Astrograph Catalog (UCAC3; Zacharias et al. 2010), with account for Star A's proper motion, μ_α = −8.9 ± 1.3 mas yr⁻¹ and μ_δ = 7.8 ± 2.2 mas yr⁻¹, given in the same catalog and independently verified by us (see the Appendix). As the difference between these and the nominal SBC coordinates (057 and −059 in right ascension and declination, respectively) is within the error budget of the HST absolute astrometry, we conclude that the SBC northwestern source is indeed Star A.

Applying the boresight correction to the SBC image, such that the coordinates of the northwestern source coincide with the UCAC3 coordinates of Star A, we obtain the following coordinates of the southeastern source

$$\begin{equation} \alpha =10^{\rm h}57^{\rm m}58\mbox{$.\!\!^{\mathrm s}$}954, \quad \delta =-52^\circ 26\mbox{$^\prime $}56\farcs 37, \end{equation} \tag{ 2 }$$

with a nominal position uncertainty of about 007. This uncertainty is mostly determined by the uncertainty of the UCAC3 absolute astrometry, thanks to the small angular distance (441) between Star A and PSR B1055−52, and the ∼01 accuracy on the HST roll angle with respect to the equatorial reference frame.

The derived position is consistent with the candidate pulsar counterpart position in the FOC observation (α = 10^h57^m58832, δ = −52°26''5628), within the uncertainty of the FOC position (≈10)⁷ and the uncertainty caused by an unknown proper motion since the epoch of the FOC observation (1996.36). This suggests that we indeed detected the PSR B1055−52 candidate counterpart of M+97 in our SBC image, while the extremely blue color (obvious from the comparison of the FUV (SBC) image with the optical (FOC and WFPC2) images; see Figure 1 and also Section 3.2) virtually proves that this object is indeed PSR B1055−52.

Using the offsets between the pulsar counterpart and Star A measured in the SBC image (202 and −392 in right ascension and declination, respectively), we looked for the pulsar counterpart at the corresponding position in the WFPC2 images. We detected it in both the F702W image (seen in the circle labeled "SBC" in the right panel of Figure 1) and the F555W image but not in the F450W.

We measured the magnitudes and fluxes of the PSR B1055−52 counterpart in the WFPC2, ACS/SBC, and FOC images through customized aperture photometry using the IRAF package digiphot. For the WFPC2 F555W and F702W bands, we used a small source aperture with radius r = 3 pixels (0135) to maximize the signal-to-noise ratio (S/N). We sampled the background in a surrounding annulus with inner radius r_b = 5 pixels and radial width δr = 10 pixels; the distance of 2 pixels from the source aperture was chosen to exclude the brighter portion of the point-spread function (PSF) wings. We then applied the aperture correction using the fractional encircled energy (FEE) coefficients (Holtzman et al. 1995), and the correction to compensate for the time- and position-dependent charge transfer efficiency (CTE) losses of the WFPC2 detector (Dolphin 2009). For estimating the 3σ flux upper limit in the F450W band, we used the same r = 3 pixels source aperture but sampled the background in a wider annulus, δr = 30 pixels, around the expected pulsar position.

For the SBC data, we chose a source aperture of r = 6 pixels (02) and a background annulus with r_b = 20 pixels and δr = 10 pixels. The large radial distance of 14 pixels between the source aperture and the background annulus was chosen to exclude the count excess over the PSF wings, which is seen from the comparison of the count distributions around the pulsar and around Star A (see the insets in the left panel of Figure 1). The 3σ excess (e.g., 114 ± 38 counts in the 03–07 annulus) cannot be ascribed to different PSFs of the pulsar and Star A because the latter is only 1 mag brighter than the pulsar in the SBC F140LP band, and it is separated from the pulsar by only 441. From these data alone, we cannot exclude the possibility that the excess is a compact pulsar wind nebula (PWN), but, as the excess is not seen in other bands, it can also be just a small-scale enhancement of the detector background at the pulsar position. Since the brightness profile of Star A is closer to that of the model PSF, we used it as a reference to compute the aperture correction by fitting its growth curve (i.e., the number of source counts within an aperture as a function of its radius).

We also measured anew the magnitude and flux of the counterpart in the FOC image. Following the approach described by Pavlov et al. (1997), we chose an optimal source aperture from the measured growth curves for each of the three orbits. We found that the growth curves for the first and third orbits were consistent with each other and compatible with the nominal values of FEE at ≈3400 Å, while the growth curve for the second orbit showed a peculiar behavior (e.g., no saturation up to r ∼ 20 pixels, which means an unusually broad PSF). Therefore, we excluded the second orbit data from the analysis and chose r = 8 pixels (011), r_b = 10 pixels, and δr = 10 pixels.

In all cases, we applied the count-rate-to-flux conversion by computing the photometric zero points in the STmag system using the image keywords PHOTFLAM and PHOTZPT, derived by the HST data reduction pipelines. To calculate the extinction-corrected fluxes, we estimated the interstellar reddening toward the pulsar, E(B −  V) = 0.07, using the hydrogen column density N_H = 2.7 × 10²⁰ cm⁻², derived by De Luca et al. (2005) from the X-ray fits, and using the correlation between N_H and E(B − V) found by Predehl & Schmitt (1995). We then calculated the extinction coefficients using the optical extinction curves by Fitzpatrick (1999) for the WFPC2 and FOC passbands and the UV extinction curves by Seaton (1979) for the SBC passband. The magnitudes and the measured, F^obs_ν, and dereddened, F^der_ν, spectral fluxes are presented in the last two columns of Table 1.

We plotted the dereddened spectral fluxes F^der_ν versus frequency ν in Figure 2. The shape of the broadband spectrum and, in particular, the relatively high FUV brightness of the object are inconsistent with it being a non-collapsed star in the Galaxy or an extragalactic object. Therefore, the photometry proves unequivocally that the source is the optical counterpart of PSR B1055−52.

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It is clear from Figure 2 that a simple one-component spectral model cannot fit the optical–UV spectral energy distribution (SED). However, similar to the optical–UV spectra of other middle-aged pulsars (e.g., KP07), the SED can be fitted by a model consisting of an (optical) power-law (PL_O) component and a Rayleigh–Jeans (RJ) component:

$$\begin{equation} F_\nu ^{\rm der} = f_{\rm O}\left(\frac{\nu }{\nu _0}\right)^{-\alpha _O} +\, \frac{2\pi }{c^2}\,\frac{R_{\rm O}^2}{d^2}\, kT_{\rm O}\,\,\nu ^2, \end{equation} \tag{ 3 }$$

where R_O is the radius of equivalent emitting sphere and T_O is the brightness temperature (both as seen by a distant observer). At E(B − V) = 0.07, the fitting parameters are α_O = 1.05 ± 0.34, f_O = 0.112 ± 0.003 μJy at ν₀ = 1 × 10¹⁵ Hz, and R²_OT_O = (510 ± 81) d²₇₅₀ km² MK (or T_O = (3.02 ± 0.48) d²₇₅₀ R⁻²_O,13 MK, where R_O,13 is the radius in units of 13 km). The PL_O component, which can be interpreted as magnetospheric radiation, dominates in the optical (λ ≳ 3000 Å). The energy flux of this component is ${\cal F}_{\rm O}^{\rm nonth} \approx 1.3 \times 10^{-15}$ erg cm⁻² s⁻¹ in the 3000–9000 Å band. The RJ component dominates in the FUV and is likely emitted from the NS surface. We will discuss implications of these results in Section 4.1.

3.3.1. Absolute Position of the Pulsar

The pulsar counterpart position given by Equation (2) was obtained from the SBC astrometry using the Star A position in the UCAC3 catalog. Of course, using just one reference star may introduce a significant bias in the absolute target position. Fortunately, the WFPC2 has a much larger FOV than the SBC, allowing astrometry with larger numbers of reference stars.

For the WFPC2 astrometry, we used two catalogs, GSC2.3 (Lasker et al. 2008) and Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006). Although the astrometric accuracy of these catalogs is somewhat lower than that of UCAC3, they contain more stars per given area, which improves the statistics of the astrometric fits. Moreover, most UCAC3 stars identified in the WFPC2 FOV are saturated, which prevents their use for the astrometric calibration. We used the F702W image, in which PSR B1055−52 was detected with a higher S/N than in the F555W image. We produced the WFPC2 mosaic image with the STSDAS task wmosaic, which also corrects for the geometric distortion. Then, we measured the centroids of the reference stars (13 stars in GSC2.3 and 16 stars in 2MASS) in the WFPC2 pixel coordinates through Gaussian fitting with the Graphical Astronomy and Image Analysis (GAIA) tool⁸ and used the catalog sky coordinates of these stars to compute the pixel-to-sky coordinate transformation with the code ASTROM.⁹ This yielded astrometric fits with the rms values σ_r = 017 for both the GSC2.3 and 2MASS stars. To these we added in quadrature the uncertainties of the registration of the WFPC2 image on the catalog reference frames, σ_tr = 014 and 008, for the GSC2.3 and 2MASS, respectively. Following Lattanzi et al. (1997), these uncertainties were estimated as σ_tr = (3/N_cat)^1/2σ_cat, where 3^1/2 accounts for three free parameters in the astrometric fit, σ_cat is the mean positional error of the catalog coordinates (03 and 02 for the GSC2.3 and 2MASS, respectively), and N_cat is the number of catalog stars used to compute the astrometric solution. Accounting for the 015 and 0015 uncertainties on the link of the GSC2.3 and 2MASS to the International Celestial Reference Frame (ICRF), the overall uncertainties of the astrometry (1σ position errors) are δ_r = 026 and 019, respectively.

We applied these astrometric solutions to compute the sky coordinates of PSR B1055−52 from its pixel coordinates. As it has been done for the reference stars, we computed the pixel coordinates through a Gaussian fitting, with an uncertainty of ⩽001 for the pulsar centroid, negligible in comparison with the overall uncertainty of our absolute astrometry calibration. We then obtained the GSC2.3-based pulsar coordinates

$$\begin{equation} \alpha =10^{\rm h} 57^{\rm m} 58\mbox{$.\!\!^{\mathrm s}$}961, \quad \delta = -52^\circ 26\mbox{$^\prime $}56\farcs 17, \end{equation} \tag{ 4 }$$

with the position error δ_r = 026, and the 2MASS-based coordinates

$$\begin{equation} \alpha =10^{\rm h} 57^{\rm m} 58\mbox{$.\!\!^{\mathrm s}$}967, \quad \delta = -52^\circ 26\mbox{$^\prime $}56\farcs 31, \end{equation} \tag{ 5 }$$

with δ_r = 019. These two positions are consistent with each other and with the UCAC3-based position derived by using Star A as the only reference star in the SBC astrometry (Equation (2)).

Since the coordinates in Equations (4) and (5) have been computed using different catalogs and different reference stars, they can be considered independent and hence can be averaged. The calculation of the weighted means yields

$$\begin{equation} \alpha =10^{\rm h} 57^{\rm m} 58\mbox{$.\!\!^{\mathrm s}$}965, \quad \delta = -52^\circ 26\mbox{$^\prime $}56\farcs 26, \end{equation} \tag{ 6 }$$

with δ_r = 015. Thus, the WFPC2 astrometry, recalibrated with the GSC2.3 and 2MASS, has provided the absolute pulsar coordinates at the epoch of 2008.18, which is separated by about 30 years from the epoch of the only published radio position.¹⁰

3.3.2. Proper Motion of the Pulsar

We used the optical coordinates of PSR B1055−52 to measure its proper motion from the comparison with the radio coordinates at the 1978.13 epoch (Equation (1)). We note that, since the GSC2.3 and 2MASS, as well as UCAC3, are linked to the ICRF, there is no a systematic offset between the optical and radio coordinates. Using Equation (6), we infer the displacement Δα cos δ = +114 ± 029 and Δδ = +004 ± 032, which corresponds to the proper motion

$$\begin{equation} \mu _{\alpha } = +38\pm 10\,\,{\rm mas\, yr}^{-1}, \quad \mu _{\delta }= +1\pm 11\,\, {\rm mas\, yr}^{-1} \end{equation} \tag{ 7 }$$

in right ascension and declination. Using the UCAC3-based positions (Equation (2)) results in virtually the same proper motion (μ_α = 35 ± 9 mas yr⁻¹, μ_δ = −2 ± 10 mas yr⁻¹), whose errors are dominated by the errors of the radio position (Equation (1)).

The pulsar proper motion can be independently measured through relative HST astrometry, by comparing its position with respect to Star A in the SBC image (epoch 2008.12) with that in the FOC image (epoch 1996.36). Of course, measuring a relative proper motion with only one reference star comes with caveats. First, Star A is slightly saturated in the FOC image and part of its PSF is outside the FOC FOV (see Figure 1 of M+97), which lowers the precision of the star centroid determination. To be very conservative, we assumed an uncertainty of 5 pixels (007). The pulsar centroids were measured with a much higher precision of ≲0.1 pixels in both the SBC and FOC images (≲00025 and ≲00014, respectively), and the residual geometric distortion (about 0004 and 0007 rms for the SBC and FOC, respectively; see Apellániz & Cox 2008 and Nota et al. 1996, respectively) is also much smaller than the FOC centroiding uncertainty for Star A.

Second, with only one reference star, we have to rely upon the nominal values of the angles between the detector axes and the directions of right ascension and declination, which may introduce an error in the direction of the proper motion. However, this error is very small in our case, thanks to the accuracy of the detector position angle (P.A.) and the small separation of Star A from the pulsar (see Section 3.1). Last but not least, the uncertainty of the proper motion of the reference star should be small enough not to hamper the measurement of the relative pulsar proper motion. We carefully verified it for Star A (see the Appendix).

Using the value of the pixel scale for the two detectors and the nominal orientations of the two images with respect to the equatorial reference frame, we found the displacement of the pulsar with respect to Star A during the time span of 11.76 years between the two HST observations: Δα cos δ = 061 ± 014 and Δδ = −019 ± 014. The uncertainty of the displacement is dominated by the large uncertainty of the Star A centroid in the FOC image. This displacement corresponds to the proper motion μ_α = 52 ± 6 mas yr⁻¹ and μ_δ = −16 ± 6 mas yr⁻¹. Correcting these values for the proper motion of Star A as given in the UCAC3 catalog (see Section 3.1), we obtained

$$\begin{equation} \mu _{\alpha } = 43\pm 6\,\, {\rm mas\, yr}^{-1}, \quad \mu _\delta = - 5 \pm 6\,\, {\rm mas\, yr}^{-1}. \end{equation} \tag{ 8 }$$

This proper motion agrees with that measured from the comparison of the radio and optical absolute coordinates. The weighted mean of the proper motion values determined by the two methods is

$$\begin{equation} \mu _\alpha = 42\pm 5\,\, {\rm mas\, yr}^{-1}, \quad \mu _\delta = -3 \pm 5\,\, {\rm mas\, yr}^{-1}, \end{equation} \tag{ 9 }$$

or

$$\begin{equation} \mu = 42\pm 5\,\, {\rm mas\, yr}^{-1}, \quad {\rm P.A.} = 94^\circ \pm 7^\circ, \end{equation} \tag{ 10 }$$

for the total proper motion and position angle (counted east of north). The large proper motion provides additional evidence that the M+07 candidate is indeed the pulsar counterpart.

We have shown in Section 3.2 that the optical–UV SED for PSR B1055−52 can be described by a sum of PL_O (magnetospheric) and RJ (thermal) spectra (see Equation (3)). It is interesting to compare these optical components with the corresponding spectral components at higher energies, particularly in soft X-rays where both the thermal and magnetospheric components are seen. As we have mentioned in Section 1, the pulsar's X-ray spectrum can be described by a model that consists of cold BB (BB_C), hot BB (BB_H), and X-ray power-law (PL_X) components. The BB_C and BB_H components are presumably emitted from the bulk of the NS surface and polar caps, respectively, while the PL_X component, $F_\nu = f_{\rm X} (E/E_0)^{-\alpha _{\rm X}}$, represents the magnetospheric X-ray emission. The parameters of the three components, as inferred from the XMM-Newton data by De Luca et al. (2005), are the following: T_C = 0.79 ±  0.03 MK, R_C = 12.3^+1.5_−0.7 d₇₅₀ km; T_H = 1.79 ±  0.06 MK, R_H = (0.46 ±  0.06) d₇₅₀ km; and α_X = 0.7 ±  0.1, f_X = 1.3^+0.2_−0.1 × 10⁻³¹ erg cm⁻² s⁻¹ Hz⁻¹ at E₀ = 1 keV, respectively. These components and their extrapolations into the optical–UV domain are shown in the left panel of Figure 3.

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We see from Figure 3 that the extension of the best-fit PL_X component into the optical overshoots the optical fluxes by a factor of ∼3, similar to most of other pulsars detected in the optical (e.g., KP07; Mignani et al. 2010). However, in contrast to the majority of those pulsars, the PL_O component is apparently steeper than the PL_X component (α_O = 1.05 ± 0.34 versus α_X = 0.7 ± 0.1), and perhaps steeper than the PL_O components in other pulsars (e.g., α_O = 0.41 ± 0.08 and 0.46 ± 0.12 in PSR B0656+14 and Geminga, respectively; see KP07). Moreover, the extrapolation of the best-fit PL_O component into the X-ray range does not intersect the PL_X spectrum, being at least a factor of ∼10 lower at E ≳ 1 keV. Such behavior is unusual (a notable exception is PSR B0540−69; Mignani et al. 2010), and it might suggest that different populations of ultrarelativistic electrons are responsible for the optical and X-ray emission of PSR B1055−52. However, taking into account the large uncertainties of the spectral slopes, it seems more plausible that α_O is substantially smaller than its best-fit value (e.g., α_O ≲ 0.5), in which case the extrapolation of the PL_O component can, in fact, be smoothly connected with the PL_X spectrum.

Interestingly, the slope α_γ = 0.06 ± 0.1 of the Fermi LAT spectrum, fit by a PL with an exponential cutoff ($F_\nu \propto E^{-\alpha _\gamma } \exp (-E/E_{\rm cut})$; Abdo et al. 2010), is even flatter than the X-ray slope. On the other hand, as we see from the right panel of Figure 3, the optical and γ-ray points can be connected by a PL spectrum with the slope α_Oγ ≈ 0.46, which, however, goes slightly above the PL tail of the X-ray spectrum. We cannot rule out the possibility that further observations (especially in the IR-optical), supplemented by a joint multiwavelength analysis, will show the magnetospheric spectrum of PSR B1055−52 to be similar to those of other pulsars with optical, X-ray, and γ-ray counterparts, including the other two Musketeers.

Another property of PSR B1055−52 to compare with those of other pulsars detected in the optical is the relationship between the optical and X-ray luminosities of the magnetospheric emission. For instance, for the wavelength range 4000–9000 Å, the flux in the PL_O component, ${\cal F}_{\rm O}^{\rm nonth} \approx 0.94\times 10^{-15}$ erg cm⁻² s⁻¹, corresponds to the luminosity $L_{\rm O}^{\rm nonth} = 4\pi d^2 {\cal F}_{\rm O}^{\rm nonth} = 6.3\times 10^{28}\, d_{750}^2$ erg s⁻¹ and optical efficiency $\eta _{\rm O}\equiv L_{\rm O}^{\rm nonth}/\dot{E}=2.1\times 10^{-6}\, d_{750}^2$. At the adopted distance of 750 pc, both the luminosity and efficiency of PSR B1055−52 are considerably higher than those of PSR B0656+14 and Geminga, for which L^nonth_O = 1.6 × 10²⁸ d²₂₉₀, η_O = 4.2 × 10⁻⁷ d²₂₉₀, and L^nonth_O = 4.8 × 10²⁷d²₂₅₀, η_O = 1.5 × 10⁻⁷d²₂₅₀, respectively (KP07). This hints that the actual distance to PSR B1055−52 is smaller, which is supported by the analysis of the thermal emission below. On the other hand, the ratio of the optical (4000–9000 Å) and X-ray (1–10 keV) nonthermal luminosities, L^nonth_O/L^nonth_X = 0.9 × 10⁻² (which does not depend on the distance), is within the same range of ∼10⁻³–10⁻² as for all other pulsars with optical counterparts (Zavlin & Pavlov 2004; Zharikov et al. 2006).

Let us now compare the optical–UV and X-ray thermal components of the PSR B1055−52 spectrum. The BB_H component is not seen in the optical because of the small emitting area, while the extrapolation of the BB_C into the optical–UV is a factor of ≈4 below the observed RJ component. Such an "optical–UV excess" of the RJ component has been seen in the nearby radio-quiet isolated NSs (RQINSs), such as RX J1856.5–3754 and RX J0720.4–3125 (e.g., Kaplan 2008 and references therein), and in the millisecond pulsar J0437−4715 (Kargaltsev et al. 2004). It, however, has not been observed in middle-aged pulsars. In particular, the extrapolated BB_C component lies slightly above the optical–UV RJ component in Geminga (Kargaltsev et al. 2005), while these components virtually coincide with each other in PSR B0656+14 (KP07).

The nature of the optical–UV excess is not fully understood; a popular hypothesis is that it is due to a nonuniform distribution of the temperature over the NS surface, such that the BB_C component originates from a smaller and hotter region than the optical–UV RJ component (e.g., Pavlov et al. 2002). Such an assumption is crudely equivalent to adding a "very cold" thermal component, BB_VC, which dominates in the UV but makes very little contribution in X-rays. In other words, the product R²_OT_O in the second term of Equation (3) is approximately equal to R²_CT_C + R²_VCT_VC. Taking into account that R²_CT_C ≈ 119 d²₇₅₀ km² MK and R²_OT_O ≈ 510 d²₇₅₀ km² MK, we obtain R²_VCT_VC ≈ 391 d²₇₅₀ km² MK (or T_VC ≈ 2.31 d²₇₅₀ R⁻²_VC,13 MK). Unfortunately, we cannot measure T_VC independent of d²/R²_VC in the RJ regime. We can, however, estimate an upper limit on T_VC from the requirement that the BB_VC spectral flux extrapolated to soft X-ray energies becomes so small, in comparison with the BB_C, that it does not affect the X-ray fit. We found that it occurs at T_VC ≲ 0.45 MK (at this temperature the contribution of the BB_VC flux is ≲10% above 0.3 keV; see Figure 3, left).

Such a limit on T_VC implies a considerably smaller distance to the pulsar than that estimated from the pulsar's dispersion measure (DM). Indeed, from the above estimate on R²_VCT_VC, we obtain d ≲ 25.4(R_VC/1 km) pc < 330 R_NS,13 pc, where we took into account that R_VC is smaller than the NS radius, R_NS = 13 R_NS,13 km. Since the NS radius (as seen from infinity) can hardly exceed 20 km, the distance is <500 pc, and it can even be substantially smaller than this upper limit.

We should note that such estimates are, of course, model dependent. For instance, the temperature distribution over the NS surface can be smooth (i.e., there are no three distinct regions as we implicitly assumed above). Moreover, the spectrum of the thermal emission from an NS surface region can differ from the BB, which may lead to an excess of the actually emitted optical–UV thermal component over the extrapolation of the X-ray thermal component even if the surface temperature is uniform (e.g., in the case of a light-element, H or He, NS atmosphere; Pavlov et al. 1996). However, fitting the model light-element atmosphere spectra to the observed X-ray spectrum always leads to a lower temperature and a larger value of R/d (i.e., to a smaller distance at a given NS radius) than those obtained from the BB fits (Pavlov et al. 1995). Therefore, it seems hard to avoid the conclusion on a smaller distance,¹¹ although its value cannot be determined accurately from the data available. We believe that 200 pc and 500 pc are reasonable lower and upper limits, and we will scale the distance to 350 pc below.

The conclusion that the distance is smaller than previously thought has important implications. For instance, the radius of the BB_C emitting region, R_C = 5.7^+0.7_−0.3 d₃₅₀ km, becomes substantially smaller than a plausible NS radius. It means that T_C = 0.79 ± 0.03 MK is not the temperature of the entire NS surface, and it should not be used for comparisons with the NS cooling models. This removes the problem of PSR B1055−52 being too hot and luminous (L_C,bol = 4.2 × 10³² d²₇₅₀ erg s⁻¹) for its age, in comparison with other middle-aged NSs and the predictions of standard NS cooling models (e.g., Yakovlev & Pethick 2004). It also means that PSR B1055−52 is not a "very slowly cooling NS" (as suggested by Kaminker et al. 2002), and there is no need to invoke an unusually low NS mass to explain its thermal emission (see Yakovlev & Pethick 2004 for references). To infer the true thermal luminosity and average surface temperature, and compare them with the predictions of the NS cooling models, the parallax-based distance to the pulsar should be measured, and the distribution of the temperature over the NS surface should be determined from the phase-resolved spectral analysis of the pulsar's soft X-ray and FUV emission.

Another implication of the smaller distance is that the magnetospheric luminosities and efficiencies in various spectral bands are lower than previously thought, being close to the luminosities and efficiencies of other pulsars. For instance, if d ∼ 350 pc, its γ-ray efficiency, η_γ ∼ 0.13 d²₃₅₀ in the 0.1–100 GeV band (Abdo et al. 2010), although still rather high, becomes not so different from the efficiencies of other pulsars detected with Fermi LAT, and the optical efficiency, η_O = 4.6 × 10⁻⁷ d²₃₅₀, becomes comparable with those of PSR B0656+14 and Geminga.

PSR B1055−52 is the third rotation-powered pulsar, after Geminga (Caraveo et al. 1998) and PSR B0540−69 (Mignani et al. 2010), for which an absolute position has been determined with a sub-arcsecond accuracy using optical astrometry techniques. Knowing the astrometric position will be useful for the precise timing analysis of the pulsar.

The proper motion we measured for the pulsar's optical counterpart (Equation (10)) corresponds to a tangential velocity V_t = (70 ± 8) d₃₅₀ km s⁻¹. This value is well below the average 400 km s⁻¹ for radio pulsars (Hobbs et al. 2005), but it is close to that of the Vela pulsar (V_t ∼ 65 km s⁻¹; Caraveo et al. 2001).

The proper motion in Galactic coordinates,

$$\begin{equation} \mu _l = 39\pm 5 \,\, {\rm mas\, yr}^{-1}, \quad \mu _b = 15\pm 5 \,\, {\rm mas\, yr}^{-1}, \end{equation} \tag{ 11 }$$

shows that the pulsar, whose current coordinates are l = 285984 and b = +6649, is moving away from the Galactic plane. We attempted to use the proper motion to locate the pulsar's birth place, identify a parent star cluster (OB association), and estimate the actual age of the pulsar that can differ from its spin-down age. However, to calculate the pulsar's trajectory back in time in the Galactic potential, one should know the present distance (which is poorly known), the tangential velocity (proportional to the distance), and the radial velocity V_r (which is unknown). Therefore, one has to calculate a large set of pulsar trajectories for various values of d and V_r. Using the code developed by Vande Putte et al. (2010), we calculated the trajectories on a grid of 21 distances (in the range of 200–1200 pc) and 21 radial velocities (−500 to +500 km s⁻¹). We also selected a sample of young (<100 Myr) clusters/associations (de Zeeuw et al. 1999; Dias et al. 2002) in a 25° radius circle around the present position of the pulsar, extrapolated back in time their motion using the same code, and looked for closest approaches of the pulsar and cluster trajectories. The results, however, turned out to be rather ambiguous because different values of d and (especially) V_r resulted in quite different candidate parent associations and pulsar ages (e.g., we found 10 candidates for the age range of 400–700 kyr), and the uncertainty of V_t propagated back in time resulted in very large uncertainties on the minimum separations. These uncertainties become even larger if one also accounts for the errors on the cluster proper motions, distances, and radial velocities. To obtain a more certain solution, we have to wait for the measurement of the pulsar's parallax and more accurate measurements of the proper motion, together with observational constraints on the pulsar's radial velocity, which could come, e.g., from the modeling of a possible bow-shock PWN created in the interstellar medium by the supersonically moving pulsar (see, e.g., Pellizza et al. 2005).