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

4. Verification of the method

In the system of a gamma ray emitting jet, many physical processes occur at the same time and a large number of quantities are needed to give a full description of the gamma ray emission. Some of the physical processes are inter-related and the corresponding quantities are inter-dependent. Further, the physical conditions may vary from jet to jet to a certain extent and the same quantity may take different values for different jets. In formulating the above beaming statistics, we have made a few assumptions to simplify the derivation of the x -distribution and successively the test with a sample of data. The validity of these assumptions are needed to be verified. Also the effects of random spreads in the physical quantities on the final x -distribution should be examined. In this section we will show that all these effect are minor compared to the beaming effect and that the x -distribution is mainly governed by the beaming effect.

4.1. Monte-Carlo Simulations of the flux correlation

To demonstrate our interpretation that the flux correlation is largely due to the co-axially Doppler beaming effect, we used Monte-Carlo simulations to generate the beamed gamma ray and radio fluxes. To make the problem simpler, we assume that the standard-candle sources are homogeneously located in an Euclidean space and thus the distribution of intrinsic fluxes obeys the well-known power law . We sample the intrinsic fluxes according to this power law and multiply them with a beaming factor. The beaming factor, as defined in Sect. 2, is a function of multi-variables which include the emission region geometry factor , emission spectral index , the Lorentz factor and the viewing angle . In the Monte-Carlo simulations, we will always treat as a random variable distributed isotropically in 3-dimensional space and set its range to . The s will be taken to be fixed values. But the or will be treated as fixed parameter in some cases and as random variable in other cases. In the following we study three typical cases.

(1). The intrinsic gamma ray and radio emissions are uncorrelated, but both are boosted with the same Lorentz factor and their spectral indices are fixed. The intrinsic fluxes of gamma rays and radio are sampled from the power law distribution , respectively. Their variation ranges are set to be two decades. cos is sampled from a uniform distribution. is set to 10.0. is taken to be 1.0 for gamma rays and 0.0 for radio. Also is taken to be 3.0 for gamma rays and 2.0 for radio. The sample size is 44 which is about the same as that of the observed one. Fig. 5 displays a sample of Monte-Carlo data which is one of the least correlated cases. A correlation study is made of the Monte-Carlo samples and the Pearson's r is used to measure the significance. The confidence level for the existence of a correlation is . This indicates that only the co-axial beaming can make two uncorrelated emissions become correlated. If a straight line is fitted to the data, the slope is about 2.0.

Fig. 5. A Monte-Carlo sample of data of gamma ray and radio flux densities. There are totally 44 datum points. The gamma ray and radio intrinsic luminosities are uncorrelated, but both are boosted with the same Lorentz factor and their spectral indices are fixed. A significant correlation is created at a confidence level .

(2). The intrinsic gamma ray and radio emissions are in exact proportion in each blazar, but the Lorentz factor varies from blazar to blazar. The energy (frequency) spectral indices are fixed. The intrinsic flux of gamma rays is sampled from the power law distribution and the radio one is obtained by scaling it. Their variation ranges are set to one decade. cos is sampled from a uniform distribution. is taken to be 1.0 for gamma rays and 0.0 for radio. We take the Lorentz factor as random variable and assume that log obeys a Gaussian distribution with a mean 1.0 and a standard deviation 0.477 for both gamma ray and radio emission. Fig. 6 displays a sample of Monte-Carlo data which comprise 44 points. Clearly, we can see that the tight correlation between the two fluxes has been preserved but the slope (on logarithm-logarithm plot) has been altered from 1.0 to about 2.0. The former slope is set by our assumption and the latter one is ruled by the difference in the beaming effects of the radio and gamma rays.

Fig. 6. A Monte-Carlo sample of data of gamma ray and radio flux densities. There are totally 44 datum points. The intrinsic gamma ray and radio emissions are in exact proportion in each blazar, but the Lorentz factor varies from blazar to blazar. The energy (frequency) spectral indices are fixed. The slope of the regression line (on logarithm-logarithm plot) has been altered from 1.0 to about 2.0.

(3). The intrinsic gamma ray and radio emissions are in exact proportion and a fixed Lorentz factor is set for both, but their energy (frequency) spectral indices are independent random variables. The indices are assumed to follow Gaussian distributions with a standard deviation 0.3. The mean is set to 1.0 for gamma rays and 0.0 for radio. The intrinsic flux of gamma rays is sampled from the power law distribution and the radio one is obtained by scaling it. Their variation ranges are set to one decade. is sampled from a uniform distribution. Fig. 7 displays a sample of Monte-Carlo 44 datum points. This sample is one of the most serverly smearing cases, and the existence of a correlation has a high confidence level by the Pearson's r. The strong correlation is retained and the slope has been changed to about 2.0. If the gamma ray index is correlated with the radio one, we may expect an even stronger correlation.

Fig. 7. A Monte-Carlo sample of data of gamma ray and radio flux densities. There are totally 44 datum points. The intrinsic gamma ray and radio emissions are in exact proportion and a fixed Lorentz factor is set for both, but their energy (frequency) spectral indices are independent random variables. The strong correlation is retained at a confidence level but the slope has been changed from 1.0 to about 2.0.

All the three cases show that the co-axial beaming effect plays the dominant role in making the correlation between gamma ray and radio emission for a reasonably large Lorentz factor (). The slope of the regression line is dictated by the differential beaming effect, i.e. the difference in gamma ray beaming and radio beaming. Therefore, we can draw the conclusion that the observed correlation between the EGRET gamma ray flux and the VLBI radio flux is largely due to the co-axial beaming effect. The dispersion in the correlation can be attributed to the spreads in the Lorentz factors and energy spectral indices, as well as in the proportionality between intrinsic gamma ray and radio luminosity.

4.2. Deviations in the x -distribution

In Sect. 2, we demonstrate that the x -distribution is a perfect power law if x is solely a function of the viewing angle while other parameters in the expression Eq. (4) remain constant. However, there are actually random spreads in these parameters and one would expect a deform in the x -distribution. The x function can be divided into two parts, one is the intrinsic flux ratio of gamma ray to radio, and the other is the Doppler beaming factor ratio. Let us look at the beaming part first. Previously, the effect of a spread in the Lorentz factor on the apparent luminosity function (a power law function of the Doppler factor) has been examined by Urry and Padovani (1991). In the presence of the beaming effect, the apparent luminosity function turns from a single power law into a double power law. These authors show that the spread does not affect the indices of the two power laws but just makes the index transition smoother. That is, the random spread deforms only the end parts of a power law distribution.

The effect of a random spread in the energy/frequency spectral indices of gamma rays and radio on the beaming part of the x -distribution is negligible. This is because the index of the x -distribution is a quotient function of the spectral indices and it is insensitive to any variation in the spectral indices. Let us show it analytically. The variation of the index of x can be derived in terms of its variable a,

where a contains the energy spectral indices as defined in Sect. 2. Observations indicate that the variation of the energy spectral index is much less than a, i.e. . Thus we have for the case . In other word, the x -distribution is not largely affected by the spreads in the energy spectral indices. Of course, we may expect a deform in the end parts of the distribution.

Now turning to the part of flux ratio, we examine the random spread in it. In Sect. 2, we simply treat the flux ratio as a constant. This is a strong assumption and thus needs justification. Physically, whether the gamma ray and radio emissions are linked is not very clear but depends on model. Nevertheless, we may put forward a scaling argument in which both emissions are roughly proportional to the accretion power in a blazar. However, the random spread in it cannot be given by the argument. Here we take an empirical approach and determine the spread magnitude from the observational data.

Let us define y to be the intrinsic flux ratio and assume that its logarithm obeys a Gaussian distribution with a standard deviation , i.e.

Here the flux units are chosen in such a way that the mean of the distribution is 0. The distribution of y can be derived from the above equation,

If we take logarithm on the both sides, a parabola appears as

which is symmetrical about a vertical line.

If we define the reciprocal of y as , then t obeys the same distribution as y. This can be derived through the relation which obeys the same Gaussian distribution as . Therefore we conclude that the distributions of y and its reciprocal are the same parabola on logarithm-logarithm plot. We will use this symmetrical property to justify either the flux ratio or the beaming ratio dominates the observed x -distribution. The logic is following. If we take the two ratios as random variables, then their product x is also a random variable. The distribution of x is solely determined by those of the ratios'. It has a simple property that it is spanned by the two sub-distributions over a wider range and has a convex shape. For example, if one of sub-distribution is a power law, the product distribution is a power law in the middle part and attached with other shapes at the end parts.

The distribution of the observed is made from our sample of data and is shown in Fig. 8. If the dispersion in y is large, the x -distribution will be dominated by the y -distribution and thus have the symmetrical property as indicated above. Otherwise, the distribution will be dominated by the power law of the beaming factor ratio. Comparison of Figs. 4 and 8 clearly show a difference between these two distributions. The slope of linear fit to the three middle points in Fig. 8 is . The three middle points are well on the power law lines in both cases and therefore the power law dominates the observed distribution. If we attribute the deform at the end parts entirely to the dispersion in y, the dispersion in logarithm is about 0.3 which corresponds a factor 2 on linear scale.

Fig. 8. The distribution of the flux density ratio of radio to gamma ray one, i.e. the reciprocal of x. Both flux densities are K-corrected. The errors given are purely statistical. The three middle points align right on a straight line and a least-square fit gives a slope .

Unless the theoretical distribution of y obeys the same power law as that of the beaming factor ratio, the distinction can always be made. It is more likely that y obeys a Gaussian than a specific power law. Although no exact emission model predicts this argument, some physical justification can be made. It is generally agreed that the gamma rays and radio are produced by different populations of relativistic electrons. If there is any physical link between these populations of electrons, it must be indirect. In the mean time, many processes are involved in the link and the accumulated dispersions naturally obeys a Gaussian.

© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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