4 Spectral energy distribution

To study the spectral energy distribution (SED) of the HRX-BL Lac objects, overall spectral indices were calculated to derive general correlations within the sample. Throughout this section the extended sample is analyzed. Ledden and O'Dell (1985) defined the overall spectral index between two frequencies:

$$\alpha_{1/2} = - \frac{\log (f_{1} / f_{2})}{\log (\nu_{1} / \nu_{2})}\cdot$$ (2)

Here f₁ and f₂ are the fluxes at two frequencies $\nu_{1}$ and $\nu_{2}$. As reference frequencies we used 1.4 GHz in the radio ( $\lambda \simeq 21$ cm), 4400 Å  in the optical ($\sim$B), and $1 ~{\rm keV}$ ( $\lambda \simeq 12.4$ Å) in the X-ray region to derive the optical-X-ray $\alpha _{\rm OX}$, the radio- X-ray $\alpha_{\rm RX}$, and the radio-optical $\alpha_{\rm RO}$ spectral index.

To compare these indices with those from the literature, shifts due to the use of different reference energies have to be taken into account. It can be shown that these shifts are small as long as the spectral shape within each band can be approximated by a single power law and the spectrum is not curved. Because the radio spectra are flat ( $\alpha_{\rm R} = 0$), the flux does not change when different reference frequencies are chosen in the radio domain. But by increasing the radio reference frequency, the $\alpha_{\rm RX}$ and $\alpha_{\rm RO}$ indices steepen. For example, if the reference frequency is changed from 1.4 to 5 GHz the radio-X-ray index changes by 6%: $\alpha_{\rm RX} (5 ~{\rm GHz},1 ~{\rm keV}) \simeq 1.06 \times \alpha_{\rm RX} (1.4 ~{\rm GHz},1 ~{\rm keV})$. If our spectral indices are compared with those using a larger X-ray reference energy, similar values for $\alpha _{\rm OX}$ and $\alpha_{\rm RX}$ are expected. Because of $f_\nu \propto \nu^{-\alpha}$, the expected flux at a higher energy is lower and the flux ratios increase. At the same time however, the frequency interval increases by about the same factor, if we assume $\alpha_{\rm E} = 1$, which is a good approximation for the mean X-ray spectral energy index of BL Lac objects The same reasoning applies for the optical region, where $\alpha_{\rm E} \mathrel{<\kern-1.0em\lower0.9ex\hbox{$\sim$ }}1$, and larger changes of the relevant indices are not expected.

overall spectral indices: $<\alpha_{\rm OX}>~ = 0.94 \pm 0.23$, $<\alpha_{\rm RX}> ~=$ $0.55 \pm 0.08$, $<\alpha_{\rm RO}>~ = 0.37 \pm 0.09$, if compared to Wolter et al. (1998), Laurent-Muehleisen et al. (1999), and Beckmann et al. (2002).

The region in the $\alpha _{\rm OX}- \alpha _{\rm RO}$ plane, which is covered by the HRX-BL Lac sample, is shown in Fig. 3. The center of the area covered by this sample is similar to that of the EMSS BL Lacs (see Padovani & Giommi 1995) though a larger range in $\alpha _{\rm OX}$ and $\alpha_{\rm RO}$ is covered.

4.1 Peak frequency

In order to get a more physical description of the spectral energy distribution of the BL Lac objects, we used a simple model to fit the synchrotron branch of the BL Lac. This has the advantage of describing the SED with one parameter (the peak frequency) instead of a set of three parameters ( $\alpha _{\rm OX}$, $\alpha_{\rm RO}$, and  $\alpha_{\rm RX}$). It has been shown by several authors that the synchrotron branch of the BL Lac SED is well approximated by a parabolic fit in the $\log \nu - \log \nu f_\nu$ plane (cf. Landau et al. 1986; Comastri et al. 1995; Sambruna et al. 1996; Fossati et al. 1998). In this way the peak position ( $\nu_{\rm peak}$), the total luminosity and the total flux of the synchrotron emission can be derived. We chose the parameterization using fluxes $\log \nu f_\nu = a \cdot (\log \nu)^2 + b \cdot \log \nu + c$. Using luminosities instead of fluxes would change the absolute constant c only, leaving the position of the peak frequency unaffected.

Figure 3: The $\alpha _{\rm OX}- \alpha _{\rm RO}$ plane covered by the HRX-BL Lac objects. The points refer to the complete sample, the triangles mark additional objects found within the course of the work. Objects with $\alpha _{\rm RO}< 0.2$ are called radio quiet.

Figure 4: Parabolic fit to the data of B2 0912+29.

If only three data points were given (one in the radio, optical, and X-ray band), the parabola was definite. When more than three data points were available (i.e. the K, H, and J near infrared measurements from the 2MASS), a $\chi^2$ minimization was used to determine best fit parameters. In principle, also the spectral slope in the X-ray band could be used to constrain the fit further. Because of the large uncertainties involved deriving these slopes from hardness ratio in the RASS, they were used mostly for consistency checks. Only in cases, where the parabolic fit resulted in peak frequencies above the highest energies observed, i.e. for objects with $\log \nu_{\rm peak}$ above 2 keV and steep X-ray spectra, the slopes were taken to account. An example for a parabolic fit is shown in Fig. 4. $\nu_{\rm peak}$ is sensitive for the $f_x/f_{\rm opt}$-relation, and is therefore strongly correlated with $\alpha _{\rm OX}$. This is shown in Fig. 5.

Figure 5: Logarithm of the peak frequency vs. $\alpha _{\rm OX}$. The relation was approximated by a polynomial of third degree. The horizontal line marks the distinction between HBLs (above the line) and IBLs used in this paper.

The relation can be approximated by a polynomial of third degree. Using an F-test, parabolic fits of higher degree gave no improvements. Thus the peak frequency can also be determined, in the case no radio data is available by applying

$$\log \nu_{\rm peak} = -3.0 \cdot \alpha_{\rm OX}^3 + 13.8 \cd... ...lpha_{\rm OX}^2 - 23.3 \cdot \alpha_{\rm OX}+ 28.5 (\pm 0.51).$$ (3)

The standard error ( $\sigma = 0.51$) is based on the deviation of the data points from the fit in Fig. 5. This result is comparable to that found by Fossati et al. (1998) when studying the dependency of $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$ on the peak frequency.

  
4.2 Correlation of luminosities with the SED

A set of physical parameters which are correlated to the peak frequency are the luminosities in the different wavelength regions. To compute luminosities for all objects, the unknown redshifts were set to z = 0.3 which is the mean value for the HRX-BL Lac sample. While the luminosities L_R in the radio, L_K in the near infrared, and L_B in the optical region are decreasing with increasing peak frequency, the situation at X-ray energies is the other way round (as reported also by e.g. Mei et al. 2002; Beckmann 1999a).

The details about the correlation analysis are listed in Table 5, including the confidence level of the correlations.

   

Table 5: Correlation of luminosity with peak frequency in the extended HRX-BL Lac sample.

region	rxy Pearson	confidence level	linear regressiona
	coefficient	of correlation
radio $(1.4 ~{\rm GHz})$	-0.23	>97%	$\log L_R = -0.09 \cdot \log \nu_{\rm peak} + 26.4$
near IR (K-band)	-0.28	${>} 95 \%^b$	$\log L_K = -0.14 \cdot \log \nu_{\rm peak} + 25.9$
optical (B-band)	-0.37	>99.9%	$\log L_B = -0.13 \cdot \log \nu_{\rm peak} + 25.1$
X-ray $(1 ~{\rm keV})$	+0.51	>99.9%	$\log L_{\rm X} = +0.19 \cdot \log \nu_{\rm peak} + 17.3$
total (radio - X-ray)	-0.12		$\log L_{\rm sync} = -0.04 \cdot \log \nu_{\rm peak} + 22.0$

^a Luminosities in $[~{\rm W/Hz}]$.
^b The lower confidence level results from the lower number of objects (52) with known K-band magnitudes.
The other correlations are using the 104 BL Lacs of the extended sample.

The total luminosity $L_{\rm sync}$ within the synchrotron branch has been derived by integrating the spectral energy distribution between the radio and the X-ray band. This is a reasonable approximation as long as the peak frequency is below 1 keV ( $\log \nu = 17.4$), but systematically underestimates $L_{\rm sync}$if the peak frequency is shifted beyond $1 ~{\rm keV}$. The relation of peak frequency with the total luminosities does not show a clear correlation.