3 Gas content - luminosity relation

3.1 Surface densities versus blue surface brightness

In the following discussion we will deal with quantities $S_{\rm HI}=M_{\rm HI}/A$, $S_{\rm H_{2}}=M_{\rm H_{2}}/A$, $S_{\rm fir}=L_{\rm fir}/A$, $S_{\rm B}=L_{\rm B}/A$and $S_{\rm R}=P_{\rm R}/A$ normalized to surface area Aof a galaxy instead of $M_{\rm HI}$, $M_{\rm H_2}$, $L_{\rm fir}$, $L_{\rm B}$ and $P_{\rm R}$, in order to avoid the size effect. The blue luminosity is a tracer of the past star formation on the time scale of Gyr, whereas the FIR luminosity is a tracer of the recent star formation on the time scale of Myr. We made use of multiple regression analysis to correct the distance effect (Malmquist bias). The corrected regression coefficients are presented in Table 3. In order to have a sufficient number of objects in each morphological group for statistics, we have included paired and interacting galaxies in one group. From Table 3, one can see that the HI and $\element{H}_{2}$ surface densities are correlated with the blue surface brightness for isolated galaxies only, and correlation between these quantities for the whole sample is completely due to isolated galaxies. We can conclude that, for isolated galaxies, both $S_{\rm HI}$ and $S_{\rm H_{2}}$ depend linearly on the past star formation activity.

 

 

Table 3: Regression coefficients between surface densities and surface brightness. The slope (S), correlation coefficient (r) and probability (p) that there is no correlation between the two variables are indicated in each box. The standard error is indicated in parentheses.

	$\log{S_{\rm B}}$	$\log{S_{\rm fir}}$	$\log{S_{\rm R}}$	$\log{S_{\rm HI}}$
$\log{S_{\rm HI}}$	S=0.97(0.18)	S=0.45(0.06)	S=0.28(0.08)
(all)	r=0.64(0.09)	r=0.76(0.08)	r=0.51(0.12)
	$p < 3\times 10^{-6}$	p <10-6	p <0.002
$\log{S_{\rm HI}}$	S=0.95(0.29)	S=0.65(0.21)	S=0.39(0.26)
(isolated)	r=0.70(0.19)	r=0.69(0.23)	r=0.50(0.35)
	p < 0.01	p < 0.01	p=0.17
$\log{S_{\rm HI}}$	S=0.35(0.59)	S=0.57(0.21)	S=0.48(0.26)
(paired+	r=0.18(0.31)	r=0.65(0.22)	r=0.50(0.27)
interacting)	p=0.59	p < 0.02	p=0.09
$\log{S_{\rm H2}}$	S=1.33(0.31)	S=0.73(0.09)	S=0.57(0.10)	S=1.06(0.14)
(all)	r=0.56(0.11)	r=0.78(0.08)	r=0.65(0.10)	r=0.77(0.10)
	p < 10-4	p < 10-6	p < 10-6	p < 10-6
$\log{S_{\rm H2}}$	S=1.23(0.35)	S=0.64(0.22)	S=0.50(0.21)	S=0.87(0.22)
(isolated)	r=0.73(0.19)	r=0.68(0.19)	r=0.64(0.23)	r=0.77(0.18)
	p < 0.005	p < 0.01	p=0.04	p < 0.002
$\log{S_{\rm H_{2}}}$	S=0.33(0.53)	S=0.48(0.21)	S=0.36(0.18)	S=0.62(0.17)
(paired+	r=0.20(0.28)	r=0.58(0.24)	r=0.43(0.21)	r=0.75(0.19)
interacting)	p=0.54	p < 0.05	p=0.06	$p < 5\times 10^{-3}$

3.2 Surface densities versus surface FIR brightness

Now we discuss the relationships between the gas surface densities and the FIR surface brightness. The corrected regression coefficients are presented in Table 3. The FIR surface brightness is well correlated with the HI and $\element{H}_{2}$ surface densities for both isolated and paired+interacting galaxies. This suggests that, with an increase in the gas surface density, the FIR surface brightness increases irrespective of the galaxy environment. Another interesting result is the non-linearity of the relation between the HI gas surface density and the FIR surface brightness. This is contrasted with the almost linear relation between the $\element{H}_{2}$ gas surface density and the FIR surface brightness. This indicates that the recent star formation activity exhibits stronger dependence on the $\element{H}_{2}$ gas phase than on the HI phase. Furthermore, recently, Wong & Blitz (2002) used the azimuthally-averaged data for seven CO-bright spiral galaxies and found that the SFR surface density exhibits a much stronger correlation with the $\element{H}_{2}$ gas surface density than with the HI gas surface density. Moreover, there exists a quasi-linear relation between the SFR and $\element{H}_{2}$ surface densities. Therefore the star-forming gas in these seven galaxies exists predominantly in the molecular form. It should be noted that the mean values of $L_{\rm fir}$, $L_{\rm B}$, $M_{\rm HI}$ and $M_{\rm H_2}$ do not differ significantly for isolated and paired+interacting Mkn galaxies.

3.3 Surface densities versus radio continuum surface brightness

It is well known that the linear relation between the radio continuum and IR emission in galaxies is a consequence of the star formation activity. It is assumed that the cosmic rays arise together with ionizing radiation during the star formation. The ionized gas emits thermal radio emission, while the cosmic ray electrons interacting with a magnetic field emit synchrotron radio emission. Thus, the radio continuum and IR emission are due to the star formation activity. On the other hand, the star formation activity is related to the gas content of a galaxy. Therefore, the study of the relationship between the radio continuum emission and the gas content of galaxies is of interest. The regression coefficients corrected for the distance effect are reported in Table 3. One can see that both phases of the gas are related to the radio continuum surface brightness, but $S_{\rm R}$ is related to $S_{\rm H_{2}}$ more strongly than to $S_{\rm HI}$. When isolated and paired+interacting galaxies are considered separately, the confidence levels of the relationships are not very high. However, it could be due to the small number of objects in each group, especially for the group of isolated galaxies (11 objects). Thus, the radio continuum emission in Markarian galaxies is related to both phases of the gas, but the relation to the molecular gas is much stronger than to the atomic one. In general, the results of Sect. 3 are in good agreement with those of Casoli et al. (1996).