3 Destruction efficiency of dust

In this paper, we discuss the variance of dust-to-gas ratio of a BCD sample in terms of the variation of $\beta _{\rm SN}$ defined in Eq. (2). We estimate $\beta _{\rm SN}$ based on McKee (1989) and LF98, while we also address their differences.

We assume that gas is divided into two components: gas in the star-forming region and that in the H I envelope. Such an envelope is generally observed around a star-forming region of a BCD (e.g., van Zee et al. 1998). We denote the gas mass fraction in the star-forming region as $X_{\rm SF}$ and that in the H I envelope as $X_{\rm HI}$ (i.e., $X_{\rm SF}+X_{\rm HI}=1$). We distinguish the two regions for the comparison with the IRAS sample (Sect. 4.1). IRAS FIR bands are sensitive to dust hotter than about 25 K. Such "warm'' dust exists in star-forming regions, not in H I envelopes. Calzetti et al. (1995) have also shown by using the IRAS sample of actively star-forming galaxies that 70% of the FIR flux comes from such a warm component of dust. Thus, dust mass derived from the IRAS observation of a BCD is considered to trace the dust in the star-forming region. This suggests that it is useful for us to consider dust contained in star-forming regions as long as we are interested in the comparison of our result with the IRAS observations.

Thus, we estimate $\tau_{\rm SN}$ and $\beta _{\rm SN}$ in star-forming regions. Gas mass accelerated to a velocity of $v_{\rm s}$ by a SN, $M_{\rm s}(v_{\rm s})$, is estimated as

$$M_{\rm s}(v_{\rm s})=6800\frac{E_{51}}{v_{\rm s7}^2}~M_\odot ,$$ (4)

where E₅₁ is energy released by a SN in units of 10⁵¹ erg and $v_{\rm s7}$ is $v_{\rm s}$ in units of 10⁷ cm s^-1 (McKee 1989). In swept ISM, dust is not fully destroyed. Thus, the fraction of destroyed dust, $\epsilon$ ($\sim$0.1; McKee 1989), should be multiplied. Since we are interested in mass swept by multiple SNe in a star-forming region, SN rate must be considered, which is denoted as $\gamma$. Then, $\tau_{\rm SN}$ is expressed as

 

$$\tau_{\rm SN} = \frac{M_{\rm g}X_{\rm SF}} {\epsilon \gamma M_{\rm s}(100~{\rm km~s}^{-1})}\cdot$$ (5)

Here, we substitute $v_{\rm s}$ with the threshold velocity for dust destruction (100 km s^-1; McKee 1989). Comparing Eqs. (2) and (5), we obtain

 

$$\beta_{\rm SN}=\epsilon M_{\rm s}(100~{\rm km~s}^{-1})~ \frac{\gamma}{\psi}~\frac{1}{X_{\rm SF}}\cdot$$ (6)

We should consider both Type Ia and II SNe in estimating $\gamma$. Type II SNe occur "soon'' after a star formation, because the progenitors of Type II SNe have lifetimes much shorter than the age of the universe. On the other hand, Type Ia SNe, whose progenitors are low-mass stars, occur about 1 Gyr after a star formation. Thus, $\gamma$ and $\beta _{\rm SN}$should be sensitive to the SFH. We express $\gamma$ as

$$\gamma =\gamma_{\rm Ia}+\gamma_{\rm II},$$ (7)

where $\gamma_{\rm Ia}$ and $\gamma_{\rm II}$ are the rate of Type Ia and Type II SNe, respectively. We note that LF98 estimated $\beta _{\rm SN}$ to be 5. But, for example, the fraction between $\gamma_{\rm Ia}$ and $\gamma_{\rm II}$ changes according to SFH of BCDs. In the next section, we examine the relation between dust-to-gas ratio and metallicity for various $\beta _{\rm SN}$, whose probable range is constrained.