4 Tools

4.1 The birthrate parameter

$\rm H_\alpha$ and NIR observations provide us with information on stellar populations with different time scales: $\sim$10⁷ yrs the former and $\sim$10¹⁰ yrs the latter. The two quantities combined give the ratio of the current SFR to the average past SFR or the birthrate parameter b, as defined by Kennicutt et al. (1994).

Following Boselli et al. (2001), we use the Near Infrared luminosity $L_{\rm H}$ as a tracer of the global mass of old stars, assuming that disk galaxies have a constant $M_{{\rm Tot}}/L_{\rm H}=4.6$ within their optical radius (Gavazzi et al. 1996c). Thus we write the adimensional parameter b as:

$$% b_{{\rm obs}} = \frac{{\rm SFR}~t_{\rm o}~(1-R)}{L_{\rm H} ~(M_{{\rm Tot}}/L_{\rm H})~DM_{{\rm cont}}}$$ (3)

where SFR is derived from the $\rm H_\alpha$ luminosity with:

$$% {\rm SFR} \left[M_\odot {\rm yr}^{-1}\right] = K_{{\rm H\alpha}} L_{{\rm H\alpha}} \left[{\rm erg~s}^{-1}\right].$$ (4)

Obviously the $\rm H_\alpha$ luminosity is deblended from the observed [NII] contribution and corrected for internal extinction as in Boselli et al. (2001). For consistency with Boselli et al. (2001) we adopt $K_{{\rm H\alpha}}= 1/1.16 \times 10^{41}$ for an IMF of slope -2.5 in the mass range 0.1-80 $M_\odot$.

$DM_{{\rm cont}}$ is the dark matter contribution at the optical radius, i.e. within the $25~{\rm mag~ arcsec}^{-2}$ B band isophote, that we assume $DM_{{\rm cont}}=$ 0.5, as in Kennicutt et al. (1994).

R=0.3 (Kennicutt et al. 1994) is the fraction of gas that stars re-injected through stellar winds into the interstellar medium during their lifetime, that we assume $t_{\rm o}$ $\sim$ 12 Gyrs.

If we assume that galaxies evolved as "closed'' systems following an exponential Star Formation History (SFH), with a characteristic decay time $\tau$ since their epoch of formation ($t_{\rm o}$), their birthrate parameter can be computed analytically (see Boselli et al. 2001) as:

$$% b_{{\rm mod}} = \frac{t_{\rm o} ~{\rm e}^{-t_{\rm o}/\tau}}{\tau (1-~{\rm e}^{-t_{\rm o}/\tau})}$$ (5)

$b_{{\rm mod}}$ can be written as a function of $L_{\rm H}$ using the relation between $\tau$ and $L_{\rm H}$ found by Boselli et al. (2001):

$$% {\rm log} \tau = -0.4 \left({\rm Log} L_{\rm H} - 12 \right) [{\rm Gyr}]$$ (6)

where

$$% {\rm Log} L_{\rm H} = 11.36 -0.4 H + 2 {\rm log} ({\rm Dist}) [L_{{\rm H}_\odot}].$$ (7)

The dependence of $b_{{\rm mod}}$ on $L_{\rm H}$ is given as a dotted line in Figs. 8-10.

Although b and $\rm H_\alpha$ EW have distinct dimensions, they are strongly correlated quantities. In fact they are operationally obtained in a similar way: b is computed by normalizing the  $\rm H_\alpha$ line intensity to the NIR continuum intensity, while the equivalent width is divided by the continuum intensity underlying the $\rm H_\alpha$ line. This is shown in Fig. 3 which can be directly compared with Fig. 4 of Kennicutt et al. (1994).

Figure 3: The relation between the birthrate parameter and the $\rm H_\alpha$ emission line equivalent width.

4.2 The global gas deficiency parameter

For galaxies in our sample we estimate the "global gas content'' $M_{{\rm gas}}=M_{{\rm HI}}+M_{{\rm H2}}+M_{{\rm He}}$.

$M_{{\rm HI}}$ is available for most (95%) targets by direct 21 cm observations (see Scodeggio & Gavazzi 1993; Hoffman et al. 1996, and references therein). The mass of molecular hydrogen can be estimated from the measurement of the CO (1-0) line emission, assuming a conversion factor (X) between this quantity and the $\rm H_{2}$ surface density. X is known to vary in the range 10²⁰ to 10²¹ [mol cm^-2 (K km s^-1)^-1] from galaxy to galaxy, according to their metallicity and UV radiation field. We adopt the empirical calibration as a function of the H band luminosity:

$$% {\rm log} X = 24.23 - 0.38\times{\rm log} L_{\rm H}$$ (8)

found by Boselli et al. (2002a). The CO (1-0) line emission is unfortunately available for 52% of the considered sample (see Boselli et al. 2002a and references therein), and it is assumed 15% of the HI content for the remaining objects (as concluded by Boselli et al. 2002a).

The contribution of He, not directly observable, is estimated as 30% of $M_{{\rm HI}}+M_{{\rm H2}}$ (see Boselli et al. 2002a).

We define the "gas deficiency'' parameter $Def_{{\rm gas}}= {\rm Log} M_{{\rm gas~ref.}} - {\rm Log} M_{{\rm gas~obs.}}$ as the logarithmic difference between $M_{{\rm gas}}$ of a reference sample of isolated galaxies and $M_{{\rm gas}}$ actually observed in individual objects (in full analogy with the definition of HI deficiency by Giovanelli & Haynes 1985). Using a procedure similar to the one adopted by Haynes and Giovanelli (1984) we find that the gas content of 72 isolated objects in the Coma Supercluster correlates with their linear optical diameter (D): ${\rm Log} M_{{\rm gas~ref}}=a+b {\rm Log}(D)$, where a and b are weak functions of the Hubble type, as listed in Table 2. $Def_{{\rm gas}}$ are listed in Col. 7 of Table 4. Histograms of the $Def_{{\rm gas}}$ parameter are given in Fig. 4 for the Coma isolated objects and for the Virgo galaxies. Isolated objects have $Def_{{\rm gas}}=0 \pm 0.18$, while Virgo galaxies have significantly positive $Def_{{\rm gas}}=0.53 \pm 0.35$.

Figure 4: Histograms of the $Def_{{\rm gas}}$ parameter for the isolated galaxies in the Coma supercluster (dashed line) and for Virgo galaxies (continuous line).