Astron. Astrophys. 325, 19-26 (1997)

3. A pilot test with the current data

The only available data for the gamma ray fluxes of blazars are those given in the Second Catalog of High Energy Gamma Ray Sources compiled by the EGRET team (Thompson et al. 1995). The relevant quantities include the gamma ray number flux above 100 MeV and the power law spectral index. Since many sources are time variable and observations at different viewing periods give different fluxes and spectral indices, we average the fluxes and indices for each source with a weighting factor being proportional to the photon number recorded in each period. Then we convert the photon number flux integrated above 100 MeV into the differential energy flux at 100 MeV, assuming a single power spectrum in the energy range 100 MeV - 10 GeV. For those sources whose spectral indices are not given in the Catalog, we assume that the differential number indices are 2.0 or the differential energy indices are 1.0 in our analysis.

We have compiled two sets of radio data for the gamma ray blazars. One is from the all-sky low resolution surveys at 5 GHz (Kühr et al. 1981; Becker, White & Edwards 1991; Gregory & Condon 1991). The relevant quantities include the radio flux density at 5 GHz and power law spectral index between 2.7 GHz and 5 GHz. The other is from the all-sky VLBI survey at 2.3 GHz (Preston et al. 1985). The relevant quantity is the "correlated" flux density defined by the authors. Cosmological redshifts are needed for the K-correction in our analysis, and the data are taken from the Quasar Catalog of Hewitt and Burbidge (1993). For those blazars without z measurements, we simply assign .

Table 1 lists our sample of data. Column 1 gives the name of the sources in terms of celestial coordinates (1950); column 2 gives the redshifts; column 3 gives the integrated gamma ray number fluxes above 100 MeV in units of ; column 4 gives the indices of differential gamma ray energy spectra; column 5 gives the radio flux densities at 5 GHz; column 6 gives the radio spectral indices between 2.7 GHz and 5 GHz; and column 7 gives the radio flux densities from the VLBI survey at 2.3 GHz. The radio flux densities are in units of Jy.

Table 1. The data sample. Column 1 - the name of the sources in terms of celestial coordinates (1950); column 2 - the redshifts; column 3 - the integrated gamma ray number fluxes above 100 MeV in units of ; column 4 - the indices of differential gamma ray energy spectra; column 5 - the radio flux densities at 5 GHz in Jy; column 6 - the radio spectral indices between 2.7 GHz and 5 GHz; and column 7 - the radio flux densities from the VLBI survey at 2.3 GHz in Jy.

We start our analysis with a correlation study of the sample of data. Fig. 1 shows the scatter plot of radio spectral index versus gamma ray spectral index . Here is defined in such a way that where is the radio frequency. There are totally 31 datum points. The Pearson's r is calculated and is found to be and the corresponding probability of confidence is . Therefore, the correlation is moderately significant. The interpretation of such a correlation is not straight forward as the radio emission is generally believed to be a superposition of synchrotron radiations emerging from different parts of an inhomogeneous source (jet) in which synchrotron self-absorption plays a role. The radio index may be understood as a measure of the degree of inhomogeneity in the emission region.

Fig. 1. The scatter plot of the index of differential energy spectra of gamma ray emission versus that of radio emission for 31 blazars detected by the EGRET instrument on board CGRO. There is a moderately significant correlation with probability .

Next, we study the correlation between radio fluxes and gamma ray fluxes. The integrated gamma ray number fluxes are converted into differential energy fluxes at 100 MeV. The K-correction is made for both the gamma ray and radio flux densities. For those gamma ray sources without z measurement, we assume that ; and for those without , we assume that . In Fig. 2, the gamma ray flux density is plotted versus the radio one of low resolution. The former is in units of and the latter is in units of Jy. There are totally 44 datum points. The Pearson's r turns out to be and the corresponding probability is . It is consistent with the result of Dondi & Ghisellini (1995). The significance of this result is moderate and it may be interpreted as evidence for a kinematical link between the gamma ray and radio emission in blazars, i.e. both of the emissions are boosted by the relativistic Doppler effect.

Fig. 2. The scatter plot of the flux density of gamma ray emission at 100 MeV versus the low resolution flux density of radio emission at 5 GHz for 44 blazars. Both fluxes are K-corrected. The gamma ray flux density is in units of ; and the radio one is in units of Jy. A moderately significant correlation is found with probability .

In Fig. 3, we plot the gamma ray flux density versus the radio one of high resolution (VLBI survey). These flux densities are also K-corrected with the above procedure. There are totally 45 datum points. The correlation is found to become stronger, with and the probability of confidence . This more significant correlation strengthens our view that the gamma ray and radio emission are kinematically linked. The radio core flux is normally a small fraction of the total and it may emerge only from a specific part of the jet. Further, it is likely that the gamma ray emission in blazars emerges from the radio core where the Lorentz factors for both emissions are equal or very close in value. The deviations from the correlation are due to the random spreads in the intrinsic luminosity ratio, spectral index and Lorentz factor. These effects will be studied with Monte-Carlo simulations in Sect. 4. We may conclude that the VLBI data are more suitable for the test of beaming statistics.

Fig. 3. The scatter plot of the flux density of gamma ray emission at 100 MeV versus the VLBI flux density of radio emission at 2.29 GHz for 45 blazars. Both flux densities are K-corrected. The gamma ray flux density is in units of ; and the radio one is in units of Jy. A more significant correlation is found with probability .

Now let us perform the test with our sample of data. The observed distribution of x is made from our sample data of 45 blazars in which the radio data are taken from VLBI surveys. For those blazars without measurement, we take 0.0 for it. Similarly, for those without , we take 1.0 for it. The distribution is shown in Fig. 4 where the data are binned into 5 intervals on logarithm-logarithm scales. The errors are purely statistical one. A least-square fit to the 5 points gives the power law index (slope) . If we remove the two end points, a least-square fit to the 3 points in the middle leads to for the index. In the next section, we will show that these three points are governed by the beaming effects while the two end points are produced by other effects. The significance of the test result is not high enough to rule out any model.

Fig. 4. The distribution of the ratio of gamma ray flux density to radio one. Both flux densities are K-corrected. The units used are the same as those in Fig. 2 and Fig. 3. The errors given are purely statistical. The three middle points align right on a straight line and a least-square fit gives a slope .

© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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