Based on the apparent similarity of the radiation properties of MSPs and ordinary pulsars, and assuming the same mechanisms to generate optical emission in both types of pulsar, we expected to detect the optical counterpart of PSR J0030+0451 at the $m\sim26$ visual magnitude level, assuming a simple scaling of $\dot{E}/d^2$ from known optical fluxes of ordinary pulsars. Our observations were deeper, but still we did not detect any reliable counterpart candidate.

To try to understand what our optical non-detection implies, we have plotted the available information about the multiwavelength spectrum of PSR J0030+0451 including radio, optical, and X-ray data in Fig. 2. For the X-ray region we have included preliminary results of recent XMM-Newton observations (Becker & Aschenbach 2002) where the pulsar was clearly detected in the 0.3-7 keV range. The detection range is shown by a horizontal bar with arrows in Fig. 2. The overall spectrum compares well with multiwavelength spectra of ordinary pulsars (e.g., Koptsevich et al. 2001), i.e., the pulsar flux is higher in the radio and fades toward the X-ray range. This can be explained by different emission mechanisms in the radio (coherent) and at shorter wavelengths (non-coherent). However, a more detailed inspection of the optical/X-ray range reveals a feature which has not been seen for ordinary pulsars. For the latter, the optical flux is usually close to an extrapolation of the nonthermal high energy tail of the X-ray emission, usually described by a power-law (PL). From Fig. 2 it is clear that this is not the case for the PSR J0030+0451.

Figure 2 shows that in the XMM-Newton range the data can be fitted equally well by three different two-component spectral models: a blackbody + power-law model (BB+PL), a broken (or curved) PL model, or a model based on two different blackbodies (BB+BB) with $N_{\rm H} \le 2.5\times10^{20}$ cm^-2. From the X-ray data alone it is difficult to discriminate between the three models, although the sharp X-ray pulse profile perhaps favors the domination of a nonthermal PL component (Becker & Aschenbach 2002). The VLT upper limits allow additional constraints using an extrapolation of the X-ray model spectra toward the optical range.

$\begin{figure} \par\includegraphics[angle=-90,width=14.3cm,clip]{h4067sp.eps}\end{figure}$

Figure 2: Radio and X-ray observations of PSR J0030+0451 with the Arecibo telescope (Lommen et al. 2000), ROSAT (Becker et al. 2000), and XMM-Newton (Becker & Aschenbach 2002), as well as VLT upper limits in the $BVR{\rm_s}$ optical bands. Solid, dashed, and dot-dashed lines show the best spectral fits of the XMM-Newton data with three alternative two component spectral models: BB+PL, broken PL, and BB+BB, respectively (BB stands for blackbody). Parameters of the fits and a mean unabsorbed integrated flux $F_{\rm X}$ , approximately the same for all the fits (Becker & Aschenbach 2002), are indicated in the plot. All fits are acceptable at the same significance level in the $\sim$ 0.3-7 keV range, indicated by a horizontal bar with arrows at the upper-left, and their $\sim$ 90% uncertainties are shown as shaded regions. The unabsorbed model spectra are extrapolated toward the optical range. Previous ROSAT data contain no spectral information and provide only an integrated flux indicated in the figure by a bold cross (Becker et al. 2000), roughly consistent with the more recent XMM-Newton results. The VLT upper limits, marked by end bars, show that any nonthermal PL component obtained from the above spectral fits of the X-ray data must be strongly suppressed in the optical range. This implies either a cutoff or a strong break in the 0.003-0.1 keV range. The purely thermal BB+BB model is consistent with our optical upper limits. Further details of this plot are discussed in Sect. 3.

From Fig. 2 it is seen that the low-energy extension of the BB+PL fit overshoots the optical flux upper limits of PSR J0030+0451 by $\sim$ 5 orders of magnitude. The broken PL extension, which implies a flatter spectrum in the soft X-ray energy band, is also 3-4 orders of magnitude higher. We have not corrected the data for interstellar extinction, which is low and does not play any significant role at such large differences. The strong suppression of the PL components in the optical range suggests that these two models either should be ruled out, or that their PL components must have a strong break or even a cutoff at a photon energy somewhere in the 0.003-0.1 keV range. Such a situation has never been observed for any ordinary pulsar. For example, the spectral index $\alpha_{\nu}$ (defined as $F_{\nu} \propto \nu^{-\alpha_{\nu}}$ ) of the young Crab pulsar changes from $\sim$ 0.5 for soft X-rays to zero in the FUV/optical/near-IR range (e.g., Sollerman et al. 2000; Sollerman & Flyckt 2002). For the relatively young Vela-pulsar (Mignani & Caraveo 2001), the middle-aged PSR B0656+14 (Koptsevich et al. 2001) and PSR B1055-52 (Pavlov et al. 2002), and even for the old ordinary pulsars PSR B0950+08 (Zharikov et al. 2002) and PSR B1929+10 (Mignani et al. 2002a), the extension of the PL X-ray component matches the optical flux, suggesting the same mechanism of nonthermal emission in the optical and X-ray ranges. It is not clear what could be the reason for such a strong change in the nonthermal spectral slope of PSR J0030+0451 from negative in X-rays to positive in the optical range. As seen from Fig. 2, the X-ray PL fits and our upper limits exclude any flat extension toward the optical range, even if we place a break point energy at the lower boundary of the XMM-Newton range $\sim$ 0.1 keV. Note that the low-frequency extensions of the PL components overshoot even the radio fluxes. This is also not typical for ordinary pulsars. One can assume that nonthermal emission is due to synchrotron radiation of relativistic particles in the magnetosphere of the pulsar. In this case we obtain for the simplest monochromatic particle distribution over the energy a spectral flux of $F_{\nu} \propto \nu^{1/3}$ in the low frequency range below a maximum frequency $\nu_{\rm m} \propto B\gamma^2$ . Here B is the magnetic field, and $\gamma$ is the gamma-factor of the emitting particles. From the spectral shape suggested by the X-ray data and optical upper limits it is natural to put $\nu_{\rm m}$ near the maximum of the X-ray spectral flux at the low boundary of the XMM-Newton range. At typical gamma-factors of primary and secondary relativistic particles in pulsar magnetospheres, i.e. $\sim$ 10⁶ and $\sim$ 10, respectively, and for the period of PSR J0030+0451 $\sim$ 4.9 ms, the same peak frequency value is predicted by a model of synchrotron emission from the pulsar light cylinder suggested by Malov (2001). For the expected synchrotron flux below the adopted $\nu_{\rm m}$ value (see Fig. 2, dotted lines) we would likely hint the optical counterpart, but we did not.

The purely thermal BB+BB spectral model is consistent with our upper limits without any additional assumptions. Its Rayleigh-Jeans tail is about 6 stellar magnitudes fainter in the optical than our upper limits, and would hardly be detectable with present telescopes. Thermal photons can be emitted by hot polar caps of the pulsar. The two-blackbody fit indicates a non-uniformity in the temperature distribution over the caps. This could be described by heat propagation over the surface of the neutron star out of a hot cap core, including neutron star atmosphere effects, as has been done in the case of the MSP J0437-4715 (Zavlin et al. 2002). An additional faint PL component is required to fit an excess over the thermal emission at high energy X-rays from PSR J0437-4715. If the same would be true for PSR J0030+0451, it could be brighter in the optical than estimated from the simple BB+BB model due to a contribution from a similar nonthermal component of magnetospheric origin. Deeper X-ray observations are probably needed to detect this component in the high energy tail of the PSR J0030+0451 spectrum. The similarity of X-ray and radio pulse profiles of this pulsar suggests that radio and X-ray peaks are in phase (Becker & Aschenbach 2002), although direct timing to confirm this has not yet been done. In the frame work of the thermal model this means that radio emission is generated close to the polar cap surface and the similarity of the pulse shapes is likely caused by the same geometry of the emitting regions.

Based on our upper limits we can constrain also the efficiency of converting spindown power of PSR J0030+0451 to optical emission. The luminosity in the B band is $L_{B}=4\pi d^2F_{B} \Delta\nu_{B} \la 6.3 \times 10^{26}~d_{230}^2$ erg s^-1, and hence the optical efficiency $\eta_{B} =L_{B} /\dot{E} \la 1.9\times 10^{-7} d_{230}^2$ . Here d₂₃₀=d/230 pc is the normalized distance to PSR J0030+0451. This upper limit is about half the efficiency of the middle-aged PSR B0656+14 (at 500 pc) and exceeds the efficiencies of the Geminga and Vela pulsars by about 1 and 2 orders of magnitude, respectively (see, e.g., Zharikov et al. 2002). It is interesting to note that the expected efficiency in the BB+BB model in Fig. 2 is 2-3 orders of magnitude lower than the upper limit derived above from our optical data, and that the efficiency in the BB+BB model is comparable to that of the Vela pulsar. The comparison of the efficiencies makes sense here, since the thermal emission from hot polar caps and the nonthermal magnetospheric emission of the Vela-pulsar, both are powered by relativistic particles produced in magnetospheres of rapidly rotating NSs. Thus, we see that the optical efficiency of the PSR J0030+0451, as derived from our and X-ray observations, is not unusually low, but is compatible with the efficiency range of ordinary pulsars detected in the optical band.

It has been shown for ordinary pulsars that spectral index appears to became steeper with pulsar age in the optical range (Koptsevich et al. 2001; Mignani & Caraveo 2001), while it flattens in gamma rays (e.g., Shearer & Golden 2002). It has been noticed also across a restricted set of young and middle-aged pulsars detected in the optical and gamma regions that the gamma-ray efficiency increases with age, while the reverse is true for the optical efficiency (Goldoni et al. 1995). This would suggest that there is a reprocessing of the gamma-photons into the optical in pulsar magnetospheres, and that it is more efficient for younger pulsars than for older ones (e.g., Shearer & Golden 2002). Thus, it would be not surprising that very old PSR J0030+0451 is fainter in the optical than we expected. However, recent optical studies of old ordinary pulsars (Zharikov et al. 2002; Mignani et al. 2002a,b) have revealed nonmonotonous behavior of the optical efficiency vs. age with a minimum at $\tau \sim 10^4$ -10⁵ yr and further increase towards higher ages $\tau \ga 10^7$ yr. Old pulsars can be actually much more efficient than the middle-aged ones and produce the optical photons with almost the same efficiency as young and energetic Crab-like pulsars. In this context low optical brightness of the MSP J0030+0451 remains puzzling.

A clue to what dominates the X-ray and optical spectrum would hopefully be found from observations of PSR J0030+0451 and other MSPs in the FUV and, in particular, in the EUV range. Even deep upper limits in these ranges would help to understand how strongly the multiwavelength emission and radiation properties of MSPs differ from those of ordinary pulsars.

3 Discussion