2 Model description

In order to consider the dust formation and destruction, we analyse the dust-to-gas ratio along with the chemical evolution model by H99. The model is based on Eales & Edmunds (1996), LF98 and Dwek (1998). In our model, we do not need to model any SFH. This has an advantage in considering the dust-to-gas ratio of BCDs since the SFH of a dwarf galaxy is generally complex (e.g., Grebel 2001) and is difficult to model. We focus especially on dust destruction, because LF98 have not fully considered it. As we see later, our model is a powerful tool to know the metallicity level where dust destruction becomes effective enough to suppress the dust-to-gas ratio.

2.1 Brief review of the model

In order to investigate dust content in a galaxy, H99 has established a set of model equations describing dust formation and destruction processes. In H99, a galaxy is treated as one zone to focus on the quantities averaged over the whole galaxy. The galaxy is assumed to be a closed system; that is, mass inflow and outflow are not considered. If the metallicity of the infalling material is zero or much lower than that of the ISM in the galaxy, the relation between dust-to-gas ratio and metallicity, with which we will be concerned in this paper, is not altered by infall (Edmunds 2001; Hirashita 2001). This is because the infall dilutes both metallicity and dust-to-gas ratio at almost the same rate. Our model does not include the effect of outflow, and this is different to LF98, in which outflow is essential to explain the observed variance of the dust-to-gas ratio in BCDs. Since Tajiri & Kamaya (2002) and Legrand et al. (2001) have suggested that outflow is not efficient for BCDs, it is worth examining a case of no outflow. Indeed, we present another clear possibility to explain the large scatter of $\cal{D}$ among BCDs later.

The model equations in H99 (see the paper for details; see also LF98) describe the evolution of total gas mass ($M_{\rm g}$), the total mass of metals (both in gas and dust phases) labeled as i(M_i; $i={\rm O}$, C, Fe, etc.), and the mass of metal i in a dust phase ( $M_{{\rm d,}~ i}$). We neglect dust growth in clouds, since Hirashita (1999a) has shown that this process in low-metallicity systems such as dwarf galaxies is much less efficient than the formation of dust around stars. Then, we adopt an instantaneous recycling approximation as in LF98 and H99 according to the formalism in Tinsley (1980): stars less massive than m_t (present turn-off mass set to be 1 $M_\odot$) live forever and the others die instantaneously.

2.2 Solution of the model

Dust-to-gas ratio and metallicity of galaxies are observationally known to correlate with each other (e.g., Issa et al. 1990). This relation has recently been used as a test for chemical evolution models including dust formation and destruction (LF98; H99; Hirashita 1999a; Edmunds 2001). The model by H99 reduces the following differential equation:

 

$${\cal Y}_i\frac{{\rm d}{\cal D}_i}{{\rm d}X_i}= f_{{\rm in},i} ({\cal R}X_i+{\cal Y}_i)-({\cal R}+\beta_{\rm SN}){\cal D}_i ,$$ (1)

where ${\cal D}_i$ is the mass fraction of an element i locked up in dust (i.e., ${\cal D}_i\equiv M_{{\rm d},~ i}/M_{\rm g}$); X_iis the mass fraction of an element i(i.e., $X_i\equiv M_i/M_{\rm g}$); $f_{{\rm in},~ i}$ quantifies what fraction of an element iejected from stars condenses into dust grains; $\beta _{\rm SN}$ is "dust destruction efficiency'', which we explain later; ${\cal R}$ is the fraction of stellar mass subsequently returned to the interstellar space, and ${\cal Y}_i$ is the mass fraction of an element i newly produced and ejected by stars. The definition of $\beta _{\rm SN}$ is as follows:

 

$$\beta_{\rm SN}=\frac{M_{\rm g}}{\tau_{\rm SN}\psi} ,$$ (2)

where $\tau_{\rm SN}$ is the timescale of dust destruction by SN shocks (Eq. (5)), and $\psi$ is the star formation rate (SFR). Hereafter, $\beta _{\rm SN}$ is called "dust destruction efficiency'', because it is inversely proportional to the timescale of dust destruction ( $\tau_{\rm SN}$). This is the most important parameter in this paper.

When all the quantities except X_i and ${\cal D}_i$ are constant in time, the analytical solution obtained by LF98 is applicable. With our notations, it is rewritten as

 

$${\cal D}_i(X_i)=\frac{b}{a}X_i+(1-{\rm e}^{-aX_i})\left(\frac{c}{a}- \frac{b}{a^2}\right) ,$$ (3)

where $a \equiv ({\cal R}+\beta_{\rm SN})/ {{\cal Y}_i}$, $b \equiv {f_{{\rm in},~ i}{\cal R}} / {{\cal Y}_i}$, and $c \equiv f_{{\rm in},~ i}$.

Here, we select oxygen as a traced element (i.e., $i={\rm O}$) according to LF98, because (i) most of the oxygen is produced by massive stars (Type II SNe and their progenitors), (ii) oxygen is one of the main constituents of dust grains, and (iii) the common tracer for the metal abundance in BCDs is an oxygen emission line. The first item (i) means that an instantaneous recycling approximation may be reasonable for the investigation of oxygen abundances, since the generation of oxygen is a massive-star-weighted phenomenon. In other words, results are insensitive to the value of m_t. Following H99, we adopt $({\cal R},~{\cal Y}_{\rm O})=(0.32,~ 7.2\times 10^{-3})$, which are consistent with the relation between dust-to-gas ratio and metallicity of nearby galaxies including our BCD sample.