4 Discussion

 

In Fig. 2 we combine our results with the available data from the optical and X-ray observations of PSR B0950+08. In the optical range we neglect the interstellar extinction, which is expected to be very low for this pulsar, $E_{B-V}\la 0.02$ (e.g., Pavlov et al. 1996). We excluded from our consideration the observations in the R band by Kurt et al. (2000) because of very low signal-to-noise ratio. In the X-ray range we show unabsorbed fluxes resulted from the BB and PL spectral fits of the ROSAT (Manning & Willmore 1994) data. The fits are rescaled to the new distance value $d=262\pm5$ pc measured by Brisken et al. (2002). Within the errors the flux $F_{\rm B}$ coincides with the value $F_{\rm F130LP}=0.051\pm0.003$ $\mu$Jy measured by Pavlov et al. (1996) with the HST/FOC in the F130LP band. The B band ( $\lambda\lambda \sim 3944{-}4952$ Å) considerably overlaps with the F130LP band ( $\lambda\lambda \sim 2310{-}4530$ Å). However, their pivot wavelengths $\lambda^0_{B}=4448$ Å and $\lambda^0_{\rm F130LP}=3750$ Å are different. This difference and close flux values in two bands suggest a flat spectrum of the object in a wide $\lambda\lambda \sim 2300{-}5000$ Å spectral range. Along with the positional coincidence, we consider this flat spectrum, which is typical for pulsar optical spectra (e.g., Koptsevich et al. 2001; Mignani & Caraveo 2001), as an additional argument in favour of the detection of the PSR B0950+08 optical counterpart.

If the detected object is the pulsar, its flux in the B band does not follow the Rayleigh-Jeans law suggested by Pavlov et al. (1996) to explain the optical-UV radiation in the only F130LP band as a low temperature thermal emission from the entire surface of a reheating/cooling NS with $R_{\infty}=13$ km. In that case in the B band we would detect twice smaller flux than measured. This is well outside the uncertainties and suggests a nonthermal origin of the pulsar emission at least in the B band.

 

 

Table 3: Parameters of the radio pulsars and the efficiencies of their optical emission $\eta _B$ in the B band.

Pulsar	$\log\tau$	d	$\log\dot{E}$	B	MB	$\log L_{B}$	$\log\eta_{B}$height 2.5ex width 0ex depth 0ex
	yr	kpc	erg s-1	mag	mag	erg s-1
Crab	3.1	2.0(1)	38.65	15.25(7)1	3.74(13)	33.23(5)	-5.42(5)
B0540-69	3.2	50+5-0.6	38.17	22.0(3)2	3.48(37)	33.47(15)	-4.7(2)
Vela	4.1	0.294+0.076-0.050a	36.84	23.7(3)3	16.4+0.5-0.8	28.3(3)	-8.5(3)
B0656+14	5.0	0.5+0.26-0.3b	34.58	25.15(13)4	16.8+2.1-1.0	28.2+0.4-0.9	-6.4+0.4-0.9
Geminga	5.5	0.153+0.059-0.034c	34.51	25.7(3)5	19.8+0.8-1.0	26.95+0.16-0.10	-7.56+0.16-0.10
B1929+10	6.5	0.331(10)d	33.59	$\geq$26.26	20.0+0.2-0.2**	27.26+0.2-0.3**	-6.3+0.2-0.3**
B0950+08	7.2	0.262(5)d	32.75	27.07(16)7	19.98(19)	26.88(8)	-5.87(8)

The B band magnitudes are from:
¹ Percival et al. (1993); ² Middleditch et al. (1987) (spectroscopic data by Hill et al. (1997) give $\sim$1 $.\!\!^{\rm m}$5 smaller magnitude); ³ Nasuti et al. (1997); ⁴ Koptsevich et al. (2001); ⁵ Bignami et al. (1993); ⁶ Pavlov et al. (1996) (3$\sigma$ upper limit); ⁷ this work. The distances d are from : ^a Caraveo et al. (2001); ^b Taylor et al. (1993), Finley et al. (1992), Anderson et al. (1993); ^c Caraveo et al. (1996); ^d Brisken et al. (2002). ^** The estimates are based on the counterpart detection in the adjacent F130LP band by Pavlov et al. (1996), see Sect. 4.
Uncertainties of M_B, L_B and $\eta _B$ include uncertainties of the optical flux and distance measurements.

If the flux in the F130LP band is still dominated by the Rayleigh-Jeans tail, we would obtain $T_{\rm BB}=(3.4\pm0.3)\times 10^5$ K, which is a factor of 4-5 higher than found by Pavlov et al. (1996), mainly owing to the change of the distance to PSR B0950+08 from 130 pc to 262 pc (see the stripe-filled BB belt crossing the F130LP band and extended to X-rays in Fig. 2). A NS with such a hot surface would produce a flux $\sim$ $(0.8{-}1.9)\times 10^{-12}$ erg cm^-2 s^-1in the ( 0.08-2.4) keV band (see a big dashed cross in Fig. 2). It is well above the value $(2.4 {-} 7.3)\times 10^{-14}$ erg cm^-2 s^-1, measured by Manning & Willmore (1994) with the ROSAT under the assumption of the BB spectrum of the detected X-ray radiation (marked by a big cross below the dashed one in Fig. 2). This means that the whole surface of PSR B0950+08 is actually much cooler and its emission cannot dominate in the F130LP band. Thermal emission from hotter, $T_{\rm BB}=(2.1 \pm0.3) \times 10^6$ K, but much smaller polar caps of the pulsar with $R_{\rm pc} \sim 40$ m, which may explain the detected X-ray radiation, can hardly be visible in the optical range also because of very small areas of the caps inferred from the BB fit of the X-ray data (see square-filled BB belt in Fig. 2).

For the above reasons it is most likely that the optical radiation of PSR B0950+08 is completely dominated by nonthermal emission produced in the magnetosphere of the rapidly rotating NS, as it is believed to be for young and well studied pulsars like the Crab and PSR B0540-69. Within large uncertainties of the available X-ray data for PSR B0950+08 the PL with the spectral index $\alpha _{\nu } \simeq 0.32$ matches both the X-ray and optical fluxes including the B and F130LP bands (straight dot-dashed line in Fig. 2). This value is consistent with what was obtained by Manning & Willmore (1994) from the analysis of the ROSAT X-ray data: $F_{\rm ROSAT}^{\rm PL} = 6.3~(+12.3, -0.34)\times 10^{-14}$ erg cm^-2 s^-1, $\alpha_{\nu}=0.9~(+1.2,-0.7)$ in (0.08-2.4) keV energy range at 68% confidence levels (see dot-dashed broken lines in Fig. 2). The PL fit of the X-ray data is more preferable since the BB fit, which is statistically also acceptable, implies by an order of magnitude smaller emitting area than it is predicted by standard models of hot polar caps at the pulsar surface (e.g., Arons 1981). The inferred spectral index differs from $\alpha_{\nu} \sim 1$ which is typical for nonthermal soft X-ray radiation of most rotation-powered NSs (e.g., Becker & Trümper 1997). However, it can be as low as 0.4 for middle-aged pulsars (Koptsevich et al. 2001) and we cannot exclude a decrease of the slope of the PSR B0950+08 spectrum towards the optical range as it is seen in the case of the Crab pulsar (e.g., Crusius-Wätzel et al. 2001). More X-ray and optical data are needed to check the spectral shape for PSR B0950+08. With new data it would be also useful to perform the BB + PL, and/or NS atmosphere + PL fits (e.g., Zavlin et al. 1996) to better constrain the parameters of the nonthermal and thermal spectral components from the pulsar magnetosphere and polar caps and to estimate their contribution to the pulsar emission in different spectral bands.

Figure 3: The optical luminosity LB ( top), spin-down power $\dot E$ ( middle) and optical efficiency in the B band $\eta _B$ ( bottom) for all radio pulsars detected in the optical range as functions of pulsar age $\tau$ (see also Table 3).

If the detected object is PSR B0950+08, the pulsar optical luminosity in the B band, assuming isotropy of the emission, is $L_{B} = 4 \pi d^2F_\nu\Delta\nu = (7.6\pm1.2)\times10^{26} d^2_{262}$ erg s^-1, where d₂₆₂=d/(262 pc) is the normalized distance. As seen from Table 3 and Fig. 3 (top panel), where we collected the data in the B band for all pulsars detected in the optical range, L_B of PSR B0950+08 is comparable with the luminosity of the middle-aged Geminga pulsar but by almost six orders of magnitude smaller than L_B of much younger and energetic Crab pulsar. These data indicate that, starting approximately from the age of Geminga, the decrease of the optical luminosity with the pulsar age becomes slower than at early stages of the pulsar evolution or even disappears completely, while the spin-down luminosity continues to decrease with approximately the same rate (Fig. 3, middle panel). This means that for unknown reasons mechanisms of the optical emission may start to be more efficient again after $\sim$10⁶ yr age. The ratio of the optical luminosity to the pulsar spin-down luminosity, $\eta_{B}=L_{B}/\dot{E}$, which can be considered as a rough characteristic of this efficiency, is plotted in Fig. 3 (bottom panel). It is seen that optical photons are produced by the old PSR B0950+08 with almost the same efficiency as by the young Crab pulsar. In this respect it is not surprising that we see some indications of the presence of a compact pulsar nebula around PSR B0950+08. Manning & Willmore (1994) discuss a compact synchrotron nebula, which is unresolved with the ROSAT, as the most likely explanation of the observed X-ray spectrum of the pulsar. The estimated sizes of the possible X-ray nebula $\sim$ $(6\hbox{$^{\prime\prime}$ }{-} 1\hbox{$^\prime$ })$ are much larger than $\sim$ $1\hbox{$^{\prime\prime}$ }$ we see in our optical images. But the difference in the X-ray and optical sizes is typical for known compact pulsar nebulae where only internal brightest structures of a nebula are visible in the optical range (e.g., Weisskopf et al. 2000).

The efficiency derived by Pavlov et al. (1996) in the F130LP band for another old, $\sim$ $3.1\times 10^{6}$ yr, PSR B1929+10 is about $(3{-}8) \times 10^{-7}d^2_{170}$. This pulsar has not yet been detected in the adjacent B band, but we can assume that its flux in B is close to that in F130LP, as it is for PSR B0950+08. Scaling by the factor $\Delta \nu_{\rm B}/\Delta \nu_{\rm F130LP} = 0.24$ and by the new radio parallax based distance d=331 pc (Brisken et al. 2002) yields $\eta_{B} \sim (2.7{-}7.3) \times 10^{-7}d^2_{331}$. With this value PSR B1929+10 occupies an intermediate position at the rising part of the dependence of $\eta_{B} (\tau)$, between Geminga and PSR B0950+08, as it is expected from its age. Although our consideration of the efficiency evolution is based on the data obtained only in the B band, it appears to be qualitatively valid for the whole optical range since the broad-band spectra of all pulsars are almost flat. Hence, optical luminosities of various pulsars should not be affected strongly by insignificant differences of their spectral slopes in this range.

Physical reasons for such a high increase of the optical efficiency at late stages of the pulsar evolution are not quite clear. We can only note that the efficiency for gamma-ray radiation, $\eta_{\gamma}=L_{\gamma}/\dot{E}$, also appears to be generally higher for older pulsars (e.g., Thompson 2000). It is difficult to estimate statistical significances of these facts and their possible correlation since the numbers of pulsars currently detected in the optical and gamma-rays are too small (about ten only). Moreover, not all known gamma-ray pulsars are detected in the optical range (and vice versa). However, the observed tendency in both ranges seems to be interesting and can hardly be ignored. It is obvious that in the both wavelength ranges the radiation is nonthermal and originates in magnetospheres of rapidly rotating NSs. Thus, the increase of the efficiencies in two very different ranges may reflect an overall increase of the magnetospheric activity with the NS spin-down. A global electrodynamic model of the pulsar magnetosphere with the activity caused by the magnetic field-aligned potential drop producing electron-positron pairs in the magnetic polar regions of the magnetosphere predicts that the efficiency should increase for high energy photons $\propto$P² (Shibata 1995). However, the optical data, particularly for young pulsars, do not follow this dependence and detailed studies of the electrodynamics and radiation processes still have to be done to explain the efficiency evolution in different spectral ranges.

Further observations of the candidates to the optical counterparts of the old pulsars PSR B0950+08 and PSR B1929+10 in different spectral bands would be very useful to resolve the efficiency problem and to better understanding the nature of the optical emission of pulsars and the evolution of this emission with pulsar age. Measurements of their proper motion and the detection of pulsations with the pulsar periods in the optical range would be most important to provide firm evidence of the pulsar nature of the detected optical objects. Simultaneous studies of the optical and X-ray pulse profile would provide stronger indications whether the optical and X-ray emissions are generated by the same physical process. Our observations of a very faint PSR B0950+08 with the Subaru show that new generation of large ground-based telescopes is very effective for these studies and could lead to a considerable increase of the number of pulsars detected in the optical range in the near future.

Acknowledgements

This work was supported in part by CONACYT projects 25454-E and 36585-E, RFBR grants 02-02-17668 and 00-07-90183. We are grateful to Y. Komiyama for the help during observations with the Subaru, and to G. Pavlov for the unpublished results on revised astrometrical referencing of the HST/FOC image of the PSR B0950+08 field. Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts. ABK is grateful to the Astronomy Department of the University of Washington for hospitality. We are also grateful to M. Richer for careful reading of this text and useful remarks, to V. Palshin for a discussion, and to the referee, J. Halpern, for drawing our attention to the incorrectness of the published ASCA data analysis of PSR B0950+08.