5 Discussion

5.1 Scenario

We propose a scenario for large variation of $\beta _{\rm SN}$ for each BCD along with intermittent SFH. The time variability of SFH on a short (<1 Gyr) timescale has been suggested observationally by Searle & Sargent (1972) and theoretically by Gerola et al. (1980). Nonlinear processes in the ISM may also cause an intermittent star formation (Ikeuchi 1988; Kamaya & Takeuchi 1997 and references therein). Thus, it is pertinent to consider the intermittent SFH of BCDs.

As shown in the following, an intermittent star formation history leads to the time variation of $\beta _{\rm SN}$, because $\gamma /\psi$ depends on time. We consider an intermittent SFH: a starburst whose SFR is $\psi_{\rm burst}$ and an inter-starburst epoch whose SFR is $\psi_{\rm inter}$. We assume that $\psi_{\rm burst}=100\psi_{\rm inter}$, for example. Such a two-orders-of magnitude variation in SFR is proposed theoretically by Gerola et al. (1980) and Kamaya & Takeuchi (1997). While the starburst is going on, we expect that almost all the SNe are Type II ( $\gamma_{\rm burst}\sim \gamma_{\rm II}$, where $\gamma_{\rm burst}$ is a typical SN rate in the bursting epoch). However, in the inter-burst epoch, Type Ia SNe can be dominated ( $\gamma_{\rm inter}\sim\gamma_{\rm Ia}$, where $\gamma_{\rm inter}$ is a typical SN rate in the inter-burst). According to the model by Bradamante et al. (1998), a given stellar population releases energy in the form of Type II and Ia SNe with a ratio of 5:1 (Bradamante et al. 1998; see their Fig. 9. This value is essentially determined by the initial mass function (IMF), and they assumed the Salpeter IMF with a stellar mass range from 0.1 to 100 $M_\odot$). Since they assumed the same energy between Type Ia and II SNe, this means that the number ratio between Type Ia and II SNe is 5:1. Therefore, we expect that $\gamma_{\rm inter}\sim\gamma_{\rm burst}/5$. The intermittent star formation finally predicts that $\gamma_{\rm inter}/\psi_{\rm inter}\sim 20\gamma_{\rm burst}/ \psi_{\rm burst}$. This means that an intermittent SFH can cause a 20-times variation in $\beta_{\rm SN}\propto\gamma /\psi$during a single star formation cycle.

Furthermore, Fig. 1 shows that the value of $\beta _{\rm SN}$ has little effect on the relation between dust-to-gas ratio and metallicity for $12+\log{\rm (O/H)} < 8$. Therefore, until the metallicity level becomes $12+\log{\rm (O/H)}\sim 8$, the relation between dust-to-gas ratio and metallicity evolves in the same way whatever the value of $\beta _{\rm SN}$ might be. On the contrary, the relation is largely affected by $\beta _{\rm SN}$ if $12+\log{\rm (O/H)}>8$. Then, we study the response of the relation between dust-to-gas ratio and metallicity to the change of $\beta _{\rm SN}$ at $12+\log{\rm (O/H)}\sim 8$, as we are interested in the intermittent SFH.

First, we shall estimate a typical metallicity increment during a single star formation epoch of the intermittent SFH. The metallicity increase during an episode of star formation, $\Delta Z$, can be estimated by $\Delta Z\sim yM_*/M_{\rm g}$, where M_* is the mass of stars formed in the episode, and y is a chemical yield. If the IMF is similar to that of the Galaxy, $y\sim Z_\odot$ (i.e., $\Delta Z\to Z_\odot$for $M_*\to M_{\rm g}$). We estimate M_* by multiplying observed SFR with a duration of an episode of a star formation activity. Assuming that the SFR is 0.1  $M_\odot~{\rm yr}^{-1}$ and that the duration is 10⁷ yr (Legrand et al. 2001), we obtain $M_*\sim 10^6~M_\odot$. With typical gas mass $M_{\rm g}\sim 10^7~M_\odot$, we obtain $\Delta Z\sim 0.1~Z_\odot$. This corresponds, for example, to the metallicity increase from $12+\log{\rm (O/H)}=8.0$ to 8.2. The model by Bradamante et al. (1998) also indicates that one episode of star formation can result in such a metallicity increment.

As shown above, the effect of intermittence can be examined by changing $\beta _{\rm SN}$. In order to examine the effect of time variation of $\beta _{\rm SN}$, thus, we calculate the relation between dust-to-gas ratio and metallicity in the following two cases:

1.

$\beta_{\rm SN}=5$ for $12+\log{\rm (O/H)}\leq 8$ and $\beta_{\rm SN}=25$ for $12+\log{\rm (O/H)}>8$

2.

$\beta_{\rm SN}=5$ for $12+\log{\rm (O/H)}\leq 8$ and $\beta _{\rm SN}=1$ for $12+\log{\rm (O/H)}>8.$

In Fig. 2, we show the result of the two calculations (two solid lines). The lower and upper branches represent the cases 1 and 2, respectively. The three dotted lines show the results as in Fig. 1. We see that when the metallicity $12+\log{\rm (O/H)}$ increases from 8.0 to 8.2, the line rapidly converges to the two dotted lines, which represent the result for constant $\beta _{\rm SN}$ (1 and 25, respectively). This rapid convergence supports the idea that the dust-to-gas ratio varies widely in response to the time-evolution of $\beta _{\rm SN}$in an episode of star formation at the metallicity level of $12+\log{\rm (O/H)}\sim 8$. Thus, it is possible to determine the variance of the dust-to-gas ratio among BCDs from the time variation of $\beta _{\rm SN}$.

Figure 2: Relation between dust-to-gas ratio and metallicity. The three dotted lines show the model results same as Fig. 1. The squares are the data points same as Fig. 1. The two solid lines represents the result of the calculations which change $\beta _{\rm SN}$(from 5 to 1 and from 5 to 25 for upper and lower lines, respectively).

5.2 Age difference

If the age of BCDs varies, the present turn-off mass of stars is changed. As a result, the returned fraction of gas (${\cal R}$), the metal yield ( ${\cal Y}_i$), and the dust supply from stars ( $f_{{\rm in},~ i}$) are effectively different among BCDs. The effect of varying ${\cal R}$ and ${\cal Y}_i$ on the relation between dust-to-gas ratio and metallicity has been examined by H99. However, the resulting relation is less sensitive to the two parameters than to $\beta _{\rm SN}$. The dependence of $f_{{\rm in},~ i}$ on the turn-off mass can be important since it largely affects the dust amount in low-metallicity systems (LF98; Hirashita 1999a).

Thus, if the BCD sample proves to have a large age variation, we should reconsider the variation in dust-to-gas ratio with a time-dependent formulation. Indeed, we cannot reject the possibility that some BCDs are much younger than the cosmic age. A metal-poor BCD SBS 0335-052 may be younger than $5\times 10^6$ yr (Vanzi et al. 2000). Recently, Hirashita et al. (2002) have succeeded in explaining the dust amount of SBS 0335-052 with a time-dependent formulation applicable to young galaxies.

   
5.3 Future observations

The time variation of $X_{\rm SF}$ also leads to the time dependence of $\beta_{\rm SF}$ (Eq. (6)), although $X_{\rm SF}$ was assumed as unity (Sect. 4.1). If the various $X_{\rm SF}$ for the BCD sample is interpreted to reflect the time evolution of $X_{\rm SF}$ in each BCD, we can suggest that the gas mass in a star-forming region should change temporally because of the mass exchange between the star-forming region and the envelope. Such a mass exchange during episodic star formation activity in BCDs is indeed suggested by e.g., Saito et al. (2000). The temporal change of $X_{\rm SF}$ is also possible if the ISM in the star-forming region is consumed for star formation and locked in stellar remnants like white dwarfs, neutron stars, and black holes.

In order to constrain $X_{\rm SF}$, we need to observe H I emission or FIR emission with angular resolution fine enough. The present typical angular resolution of 1' corresponds to 2.9 kpc in physical size if a galaxy lies at a typical distance of 10 Mpc. Future large space FIR telescopes such as Herschel  (e.g., Pilbratt 2000) or SPICA (e.g., Nakagawa et al. 2000) will resolve the star-forming regions of the BCDs. For example, the Japanese future infrared satellite SPICA will have a diameter larger than 3.5 m. If the diffraction limit is achieved, the angular resolution becomes 6''at the wavelength of 100 $\mu$m. This corresponds to 290 pc at the distance of 10 Mpc, and is comparable to or smaller than the half-light radius of a typical BCD (Marlowe et al. 1999).