3 Results and discussion

In this paper we obtained a large database of flux density measurements over a wide range of frequencies, from 39 MHz to 43 GHz. Although measurements were made in three different observatories, the data seem consistent. In particular, the values obtained by Lorimer et al. (1995) and those from the Effelsberg radio-telescope at the same frequency are comparable (e.g., PSR B0740-28). We have calculated the spectral index for pulsars using the method described in Sect. 2.3. The results of this analysis are listed in Table 2. We found only 15 pulsars out of 167 whose spectral fit evidently required the two-power-law model (Table 3). In Fig. 1a we present distribution of spectral indices $\alpha$ for pulsars with a simple power-law spectrum and in Figs. 1b and c for pulsars with a broken-type spectrum (see caption for explanation of $\alpha _1$ and $\alpha _2$). In Figs. 2b and c we also show two examples of two-power-law spectrum pulsars with the smallest and largest slope difference, respectively.

As detailed above, in order to reduce the effects of diffractive and refractive interstellar scintillations as well as possible intrinsic phenomena, we need a large number of measurements at a given frequency to obtain reliable pulsar spectra. We believe that our large sample of flux density measurements is capable of doing this over a wide frequency range, allowing an analysis of the spectral behaviour of pulsar radio emission. Our analysis shows that, in principle, pulsar spectra are described by a simple power law with the mean spectral index $<\alpha>\,\,=-1.8~\pm~0.2$ (see Fig. 2a). We examined the data for any possible correlations between spectral index and rotation period P, spin-down rate $\dot P$, characteristic age $\tau$, polarization as well as profile type. In general no significant correlations were found but we have distinguished some interesting groups of objects which are discussed below:

(i) Very steep spectrum sources. This group of pulsars consists of objects with very steep spectra. Examples of such pulsars are the PSRs B0942-13, B0943+10 and B1859+03¹ with spectral indices of -3.0, -3.7 and -2.8, respectively (see Table 2). Lorimer et al. (1995) suggested that older pulsars ( $\tau \geq 10^8$yr) have steeper spectra, which is obviously not the case for B0943+10 and B1859+03 as these pulsars have characteristic ages of 4.9 10⁶ yr and 1.4 10⁶ yr, respectively. There is, of course, yet another exception, the Crab pulsar (PSR B0531+21), i.e. the youngest known radio pulsar with one of the steepest spectrum in our sample. These results provide evidence that there is no correlation between steepness of spectra and the characteristic pulsar age;

Figure 1: a) The distribution of spectral index $\alpha$ for simple power law spectra, b) The distribution of spectral index $\alpha _1$ for the low frequency part of two-power-law spectra, c) The distribution of spectral index $\alpha _2$ for the high frequency part of two-power-law spectra

(ii) Flat spectrum sources. It was previously believed that pulsars have a steep spectra but the analysis of a large sample shows that there are pulsars with flat spectra over a wide frequency range. In this group there are pulsars which have almost flat spectra with $\alpha \geq -1.0$. Examples of such pulsars are B0144+59, B1750-24 and B2022+50¹ with spectral indices of $-1.0 \pm 0.04$, $-1.0 \pm 0.07$ and $-0.8 \pm 0.05$, respectively. Although Lorimer et al. (1995) suggested that younger pulsars have flat spectra, they also found that PSR B1952+29 possessed a flat spectrum and yet had a characteristic age of 3.4 10⁹ yr. This pulsar indeed has a flat spectrum in the frequency range from 400 MHz to 1.4 GHz but considering the whole frequency range from 400 MHz to 10.7 GHz its spectrum becomes a two-power-law one. A similar behaviour was observed for PSR B0540+23¹. It is possible that the pulsars with flat spectrum mentioned by Lorimer et al. (1995) may have a break in their spectrum at higher frequencies;

Figure 2: a) Example of a typical spectrum with $\alpha =-1.8$, b) Example of the two-power law spectrum with the smallest difference in slopes ( $\alpha _1=-0.7$ and $\alpha _2=-1.2$), c) Example of the two-power law spectrum with the largest difference in slopes ( $\alpha _1=-0.6$ and $\alpha _2=-2.7$). Squares represent the measurements from Pushchino Radio Astronomical Observatory, diamonds represent measurements from Jodrell Bank and circles represent measurements from the Effelsberg Radiotelescope. Filled circles represent single measurements

(iii) Sources with low-frequency turn-over. Spectra of many pulsars show a low-frequency turn-over at $\sim$100 MHz (Sieber 1973; Izvekova et al. 1981). We have not fitted the turn-over points because of the gap in flux density measurements at frequencies between 100 MHz and 300 MHz and difficulties in determining the maximum frequency $\nu_{{\rm max}}$. We have found 2 pulsars in our sample which have a turn-over at unusually high frequency ($\sim$ 1 GHz): B1838-04 and B1823-13 (see Fig. 3). These are young pulsars and all belong to the 1800-21-class of pulsars, which was introduced by von Hoensbroech et al. (1998) to describe their unusual polarization properties. The PSR B1800-21¹ has a two power law spectrum, which may be also interpreted as a "broad form of turn-over'';

(iv) Sources with high-frequency turn-up or flattening. There is a group of pulsars which have a possible turn-up or flattening in spectra at very high frequencies. Therefore we have not fitted the points above 23 GHz. In this group there are pulsars such as: B0329+54, B0355+54, B1929+10 and B2021+51¹ which were studied in detail by Kramer et al. (1996). There are also pulsars which may show a spectrum flattening already at $\sim$ 5 GHz. For example, the spectra of PSR B0144+59 and B2255+58¹. Recently, the idea of a spectral change at very high frequencies received a strong support from observations of the Crab pulsar. Moffet & Hankins (1999) observed a clear flattening of its spectrum at realtively low frequency around 10 GHz, as compared with about 20 GHz (Kramer et al. 1996);

(v) Sources with two power law spectra. We also recognized broken-type spectra (Sieber 1973; Malofeev et al. 1994), although there are only 15 definite two-power-law cases in our sample, showing so called break frequency between 0.9 and 2.7 GHz (see Table 3) which divides the whole spectrum into two parts with considerably different slopes. This is a significantly smaller fraction (only about 10%) than the reported 35% by Malofeev (1996). The distribution of spectral indices for broken-type pulsars is shown in Figs. 1b and c. The reduced fraction of two-power-law spectra pulsars can be only partly explained by selection effects possibly present in the Malofeev sample. In fact, many pulsars which were previously thought to demonstrate two-power-law spectra (Malofeev et al. 1994; Kramer et al. 1996; Xilouris et al. 1996) can be modelled by a simple power-law spectra in our sample. We believe that this can be largely explained by severe flux density variations (see Sect. 2.2) and the fact that the number of measurements included into a fit so far may have been too small (e.g. PSR B0628+28¹).

Figure 3: a) Typical low frequency turn-over, b) an example of the unusual turn-over at around 1 GHz

We note that Gil et al. (1994) and Malofeev (1996) suggested that PSR B1822-09 has a complex flux density spectrum. Our data do not confirm such a behaviour although some deviations from a simple power law seem to be present. Moreover, there are indeed some other pulsars which exhibit a rather unusual spectral behaviour (e.g. B0823+26, B0621-04, B0656+14¹). These objects may actually have a complex spectrum but we certainly need more and better data before we should consider them as a separate class of sources. Generally, pulsar spectra can be classified in two groups: out of 167 pulsars which have been studied over a wide frequency range (from 400 MHz up to at least 5 GHz), the spectra of most pulsars can be modelled by a simple power law. About 10% of all pulsars require two power laws to fit the data. Table 4 and references therein clearly indicate that the spectra of slowly and fast rotating pulsars (i.e. millisecond pulsars) are indeed identical on the average. We note again, that the analysis of the main physical parameters of pulsars with unusual or two-power-law spectra has not shown any correlation, consistent with the results of Xilouris et al. (1996) and Malofeev (1996).