5 Distribution in space

5.1 V_e/V_a-test

Redshifts are available for 64 (83%) of the 77 BL Lac objects which form the complete sample. Therefore it is possible to determine a luminosity function for the HRX-BL Lacs and to study the evolution by application of an $V_{\rm e}/V_{\rm a}$-test. The direct images of the BL Lacs without redshift determination show point-like structure, and most of them have optical spectra consistent with high redshifts (z>0.5). The difficulty in determining redshifts for them indicate that these objects are highly core dominated with the host galaxy outshined by the BL Lac core. This implies that the optical luminosity of these objects should be quite high.

The $V_{\rm e}/V_{\rm a}$-test is a simple method developed by Avni & Bahcall (1980) based on the $V/V_{\rm max}$ test of Schmidt (1968). $V_{\rm e}$ stands for the volume, which is enclosed by the object, and $V_{\rm a}$ is the accessible volume, in which the object could have been found (e.g. due to a flux limit of a survey). Avni & Bahcall showed that different survey areas with different flux limits in various energy bands can be combined by the $V_{\rm e}/V_{\rm a}$-test. In the case of no evolution $\langle V_{\rm e}/V_{\rm a} \rangle = 0.5$ is expected and following Avni & Bahcall (1980) the error $\sigma_m(n)$ for a given mean value $\langle m \rangle = \langle V_{\rm e}/V_{\rm a} \rangle$ based on n objects is:

$$\sigma_m(n) = \sqrt{\frac{1/3 - \langle m \rangle + \langle m \rangle^2}{n}}\cdot$$ (4)

We computed the accessible volume V_a,i for each object by applying the survey limits. In most cases this volume is determined by the X-ray flux limit, only $\sim$10% of the objects have a smaller V_a,i for the radio data, due to the radio flux limit of $2.5 ~{\rm mJy}$.

Applied to the complete sample the test yields $\langle V_{\rm e}/V_{\rm a} \rangle = 0.42 \pm 0.04$. This result shows that HBLs have been less numerous and/or less luminous in the past, but the significance is only $2 \sigma$. The negative evolution of X-ray selected BL Lac objects has been reported several times before. We also performed a K-S test in order to determine the probability of uniform $V_{\rm e}/V_{\rm a}$ distribution, which would mean no evolution. For the whole HRX-BL Lac sample the probability of no evolution is rather small (3.5%).

Thanks to the large number of objects with known redshifts within the HRX-BL Lac sample it is possible to examine dependencies of the evolution on other parameters, like the overall spectral indices. A division into two groups (more and less X-ray dominated objects) according to $\alpha _{\rm OX}$ was already made by Bade et al. (1998) for the core sample and resulted in a lower $\langle V_{\rm e}/V_{\rm a} \rangle=0.34\pm0.06$ for the HBLs ( $\alpha_{\rm OX}< 0.9$) than for the IBLs within the sample. The $\langle V_{\rm e}/V_{\rm a} \rangle =0.48\pm0.08$ for IBLs was even consistent with no evolution. Dividing the HRX-BL Lac sample accordingly we now get for the HBLs ( $\alpha_{\rm OX}< 0.9$) $\langle V_{\rm e}/V_{\rm a} \rangle = 0.45 \pm 0.05$(N=34) and for the IBLs $\langle V_{\rm e}/V_{\rm a} \rangle = 0.40 \pm 0.06$(N=30). The difference between the two groups has practically vanished, and we are thus not able to confirm the different types of evolution for the HBLs and the IBLs. But still there are 13 objects within the HRX-BL Lac sample without known redshift, and nearly all of them are IBLs. Including them into the $V_{\rm e}/V_{\rm a}$-test by assigning them either the mean redshift of our sample (z=0.3) or a high redshift (z=0.7) does not change the mean $V_{\rm e}/V_{\rm a}$values significantly. The results of the different $V_{\rm e}/V_{\rm a}$-tests are shown in Table 6. Assigning even higher redshifts would increase the $V_{\rm e}/V_{\rm a}$ for the IBLs, but we consider this unlikely, as the luminosities would then become exceptionally high. For example in 0716+714, PG 1246+586, or PG 1437+398 the X-ray luminosities would exceed values of $L_{\rm X} = 10^{46} \; \rm erg \; s^{-1}$ in the $0.5 {-} 2.0 \; \rm keV$ range.

   

Table 6: Results from the $V_{\rm e}/V_{\rm a}$-tests for the HRX-BL Lac complete sample.

selection	unknown z	Na	$\langle V_{\rm e}/V_{\rm a} \rangle$	$K-S^b ~ [\%]$
	set to
all (known z)	-	64	$0.42 \pm 0.04$	3.5
all	0.3	77	$0.44 \pm 0.03$	5.3
all	0.7	77	$0.46 \pm 0.03$	5.3
HBLs (known z)	-	34	$0.45 \pm 0.05$	24.0
all HBLs	0.3	36	$0.48 \pm 0.05$	46.1
all HBLs	0.7	36	$0.48 \pm 0.05$	46.1
IBLs (known z)	-	30	$0.40 \pm 0.06$	14.0
all IBLs	0.3	41	$0.41 \pm 0.05$	10.7
all IBLs	0.7	41	$0.43 \pm 0.05$	10.7

^a Number of objects used for this test.
^b K-S test probability that the $V_{\rm e}/V_{\rm a}$ values have a uniform distribution in the [0...1] interval (probability for no evolution).

   

Table 7: Results from the $V_{\rm e}/V_{\rm a}$-tests for comparable investigations.

survey	selection	unknown z	Na	$\langle V_{\rm e}/V_{\rm a} \rangle$
REX	total	0.27	55	$0.48 \pm 0.04$
REX	HBL	0.27	22	$0.49 \pm 0.06$
sedentary	total	0.25	155	$0.42 \pm 0.02$
DXRBS	all BL Lacs	0.40	30	$0.57 \pm 0.05$
DXRBS	HBL	0.40	11	$0.65 \pm 0.09$
DXRBS	LBL	0.40	19	$0.52 \pm 0.07$

^a Number of objects used for this test.

We conclude therefore that the HRX sample shows no difference in evolution for HBLs and IBLs. The results presented here are in good agreement with recent other investigations on the evolutionary behaviour of BL Lac objects, as shown in Table 6. Except the sedentary survey (Giommi et al. 1999) none of the studies could confirm the highly significant negative evolution found e.g. by Bade et al. (1998) for the HRX-BL Lac core sample or by Wolter et al. (1994) for the EMSS BL Lacs. The best sample to be compared with should be the REX survey, which also uses the combination of RASS and NVSS data, although going to lower X-ray flux limits while using only the are of the PSPC pointed observation. The REX has also a mean redshift of z = 0.3 and the $\langle V_{\rm e}/V_{\rm a} \rangle$ are within one sigma when compared to the HRX-BL Lac sample.

5.2 X-ray luminosity functions

For the $V_{\rm e}/V_{\rm a}$-test the knowledge of the redshifts is of minor importance. However, the lumminosity function which defines the space density at a given object luminosity can only be derived when having a complete sample with known distances. Based on this function we wish to estimate the fraction of AGN which appear to be BL Lac objects. To determine the cumulative luminosity function (CLF), one has to count all objects within a complete sample above a given luminosity, and divide this number by the volume $V_{\rm a}$ which has been surveyed for these objects. We follow here the procedure described by Marshall (1985) to derive the space density and the corresponding errors. The survey area of the HRX-BL Lac complete sample is $4768 ~{\rm deg}^2$ (Table 1). Because the fraction of objects without known redshift is 17% the effective area which is used to compute the luminosity function is decreased by this fraction to $3959 ~{\rm deg}^2$. This implies that the redshift distribution of the missing objects is the same as for the rest. As discussed before, this assumption might be incorrect, as we expect many of them having rather high redshifts. The effect of different evolution for high and low redshift objects, described in the next paragraph, would be even stronger in this case.

The complete sample is large enough to divide it into a high redshift and a low redshift bin in order to examine possible differences in their CLF. The dividing value was set to the median of the HRX-BL Lac sample $z_{\rm median} = 0.272$. To derive high and low redshift CLFs the accessible volume V_a,i for the objects with z < 0.272has been restricted to z = 0.272 whenever $z_{\max,i} > 0.272$. For the high redshift objects the accessible volume was computed from z = 0.272 up to $z_{\max,i}$. The resulting two cumulative luminosity functions are shown in Fig. 6.

Figure 6: Cumulative luminosity function of the two subsamples with z > 0.272 (circles) and z < 0.272 (open triangles).

There seem to be differences between the high and low redshift CLF. The slope of the low redshift CLF is flatter. A linear regression gives a slope of -0.9 while for z > 0.272 the slope is -1.4. But in the overlapping regime at $L_{\rm X}(0.5 {-} 2.0 ~{\rm keV}) \sim 10^{45}$ the luminosity functions show similar slopes.

The left panel of Fig. 7 shows the comparison of the HRX-BL Lac complete sample X-ray luminosity function with the results from the EMSS BL Lac sample (Wolter et al. 1994; Padovani & Giommi 1995). The expected luminosities of the HRX-BL Lacs within the EINSTEIN IPC energy band ( $0.3{ -} 3.5 ~{\rm keV}$) were calculated assuming a spectral slope of $\alpha_{\rm X} = 1.0$. Space densities are given as number of objects per $\rm Gpc^{3}$ and X-ray luminosity bin following Padovani & Giommi (1995). The data from the EMSS are consistent with those from the HRX-BL Lac complete sample within the $1 \sigma$ error bars. The marginal differences can be due to systematic errors for the calculated luminosities in the IPC band because of differing spectral slopes, or resulting from differences in the calibration of the IPC and the PSPC detectors.

In the right panel of Fig. 7 we compare the differential luminosity function of the complete sample with the corresponding function for AGNs at z<0.5. The AGN X-ray luminosity function was taken from the ROSAC sample ("A ROSAT based Search for AGN-Clusters'', Tesch 2000). This AGN sample was constructed similarly as the HRX-BL Lac sample and both samples match closely in brightnesses and redshifts. The ROSAC-AGN sample contains 182 RASS-AGNs with z < 0.5 identified in an area of $363 ~{\rm deg}^2$in the constellation of Ursa Major. The AGN X-ray luminosities have been corrected for the different X-ray band ( $0.1 {-} 2.4 ~{\rm keV}$ instead $0.5 {-} 2.0 ~{\rm keV}$) using the same spectral slopes used for the ROSAC sample.

We find that the space density of BL Lacs in the luminosity range $44 < \log~L_{\rm X} < 46$ is about 10% of the space density of AGNs. In case that all AGNs have jets and would be classified as BL Lacs when looking into their jet, an jet opening angle of ${\sim} 50 ^{\circ}$ would follow. But as the jet emission is expected to be beamed, the BL Lacs appear to be brighter than they are. Following Urry & Shaefer (1984) the observed luminosity is $L_{\rm obs} = \delta^p ~ L_{\rm emi}$ with $L_{\rm emi}$being the emitted luminosity, and

$$\delta = \frac{1}{\gamma ( 1- \frac{v}{c_0} \cos \theta)}$$ (5)

where $\gamma$ is the Lorentz factor of the jet emission. p depends on the spectral slope and the jet flow model. For the simple case of a moving blob and continuous reacceleration (Lind & Blandford 1985) which applies e.g. for the model of a wide X-ray jet (Celotti et al. 1993) the exponent is $p = 3 + \alpha$, where $\alpha \simeq 1$ is the spectral index. For a conical jet this exponent is $p = 2 + \alpha$ (Urry & Shaefer 1984). Assuming an jet opening angle of $\theta \sim 30 ^{\circ}$ (Urry & Padovani 1995) and a Lorentz factor $\gamma \sim 5$ the amplification factor is $\delta^p \simeq 50$ (for the conical jet) and $\delta^p \simeq 200$ (for the wide X-ray jet). Correcting the luminosities accordingly yields a fraction of $\le$0.1% BL Lacs among all AGNs. A smaller opening angle and/or larger Lorentz factor would lead to an even lower BL Lac fraction among the AGN.

Figure 7: Left panel: The differential X-ray luminosity function of the HRX-BL Lac complete sample (circles) in comparison to EMSS BL Lacs (triangles; Padovani & Giommi 1995). The X-ray data of the HRX-BL Lac objects have been extrapolated to the EINSTEIN IPC energy band assuming a spectral slope of $\alpha _{\rm X} = 1$. Right panel: Comparison of the X-ray luminosity function of RASS selected AGNs from the ROSAC sample (triangles; Tesch 2000) with HRX-BL Lacs (circles). The density of BL Lacs is $\sim$10 times lower than for all AGNs.