4 Sample and result

   
4.1 Observational samples

We compare the model calculation in Eq. (3) with the observed values of dust-to-gas ratio and metallicity of BCDs. We adopt the data in Table 2 of LF98 and Table 1 of Sage et al. (1992). The latter sample has been adopted by Tajiri & Kamaya (2002). We select BCDs whose 21-cm H I emission and FIR dust emission are both detected. The data are summarised in Table 1.

 

 

Table 1: Data of blue compact dwarf sample.

Object	D	S60	S100	$\log M_{\rm HI}$	$\log M_{\rm d}^{\it IRAS}$	$12+({\rm O/H})$	${\rm ref.^a}$
(Mpc)	(Jy)	(Jy)	$[M_\odot]$	$[M_\odot]$
II Zw 40	9.1	6.6	5.8	8.30	4.55	8.15	S92
Haro 2	20.3	4.8	5.5	8.68	5.39	8.4	S92
Haro 3	13.7	5.2	6.7	8.76	5.21	8.3	S92
UM 439	12.6	0.39	1.2	8.23	4.96	7.98	S92
UM 462	11.9	0.99	1.1	8.15	4.21	7.89	S92
UM 465	13.2	0.99	1.3	7.71	4.48	8.9	S92
UM 533	10.4	0.51	0.54	7.76	3.75	8.10	S92
UM 448	6.00	4.14	4.32	7.54	4.16	8.08	LF98
IC 3258	21.2	0.490	0.970	8.55	5.03	8.44	LF98
Mrk 7	42.3	0.480	0.970	9.56	5.64	8.54	LF98
Mrk 33	21.6	4.68	5.30	8.77	5.41	8.40	LF98
Mrk 35	14.5	4.95	6.74	8.73	5.29	8.30	LF98
Mrk 450	12.1	0.480	0.820	7.77	4.37	8.21	LF98
NGC 4670	12.1	2.63	4.47	8.52	5.10	8.30	LF98
NGC 4861	12.9	1.97	2.26	9.13	4.60	8.08	LF98
II Zw 70	17.6	0.710	1.24	8.56	4.89	8.11	LF98

$^{{\rm a}}$

S92 - Sage et al. (1992); LF98 - Lisenfeld & Ferrara (1998).

The observational dust-to-gas ratio in a star-forming region should be

$${\cal D}^{\rm obs}\equiv M_{\rm d}^{\it IRAS}/(M_{\rm HI}X_{\rm SF}) ,$$ (8)

where $M_{\rm d}^{\it IRAS}$ and $M_{\rm HI}$ are dust mass determined from the IRAS observation and the total H I gas determined from the 21-cm observations. $M_{\rm HI}$ traces H I gas in both the star-forming region and the envelope of a galaxy. On the other hand, $M_{\rm d}^{\it IRAS}$ traces the amount of dust in the star-forming region, because the IRAS is sensitive to high-temperature ($\ga$25 K) dust. In the following two paragraphs, we describe the two quantities further.

LF98 divided the observed H I mass by a factor of 2 to obtain the gas mass in only the star-forming region. In other words, LF98 assumed that $X_{\rm SF}$ is 0.5. However, $X_{\rm SF}$ is hardly constrained observationally. In this paper we assume $X_{\rm SF}=1$for the observational sample to obtain the first result. As we discuss in Sect. 5, the variation of $X_{\rm SF}$also contributes to the variation of $\beta _{\rm SN}$. Therefore, $X_{\rm SF}$ affects our analysis in the following two ways:

1.

Observationally, the estimate of dust-to-gas ratio in a star-forming region is affected by the value of $X_{\rm SF}$.

2.

Theoretically, the value of $X_{\rm SF}$ affects the value of $\beta _{\rm SN}$, thus changing the dust-to-gas ratio.

It is difficult to discuss the former quantitatively, because we need a high-resolution spatial map of H I distribution and a reasonable observational definition of a star-forming region on the map. However, we can discuss the latter because the effect of $X_{\rm SF}$ on $\beta _{\rm SN}$ is well determined from Eq. (6).

The observational dust mass, $M_{\rm d}^{IRAS}$, is derived from the luminosity densities at wavelengths of 60 $\mu$m and 100 $\mu$m observed by IRAS using Eq. (4) of LF98. The IRAS bands are insensitive to the cold ($\la$20 K) dust that lies out of star-forming regions. Moreover, since the dust in a star-forming region suffers destruction by SN shocks, we should take into account an efficient destruction in the star-forming region. Therefore, in order to discuss the IRAS sample and the selective dust destruction in star-forming regions, we need to define $\beta _{\rm SN}$ as the dust destruction efficiency (Sect. 3) in star-forming regions as we have done in Eq. (6).

Finally, we hypothesise that the fraction of oxygen contained in dust grains is constant for all the sample BCDs. Following H99, we assume the Galactic composition of the grains:

 

$${\cal D}=2.2{\cal D}_{\rm O}.$$ (9)

   
4.2 Conclusion from our model

In Fig. 1, we show analytical results calculated according to Eqs. (3) and (9) with $i={\rm O}$ for various values of $\beta _{\rm SN}$ as solid line ( $\beta _{\rm SN}=1$), dotted line ( $\beta_{\rm SN}=5$; the case of LF98) and dashed line ( $\beta_{\rm SN}=25$), respectively. The black and gray squares indicate the observational samples in Sage et al. (1992) and LF98, respectively. The number ratio of oxygen atoms to the hydrogen atoms is denoted as (O/H). We convert the mass fraction of oxygen, $X_{\rm O}$, to (O/H) for the model prediction, assuming $\log X_{\rm O}+10.80=12+\log{\rm (O/H)}$. In this figure, $f_{\rm in,~ O}=0.1$ is assumed to concentrate on the variation in $\beta _{\rm SN}$. LF98 have shown a large variety of $f_{\rm in,~ O}$to explain the large scatter of the relation between $\cal{D}$ and $X_{\rm O}$ in their Fig. 7. Since Hirashita (1999a) has shown that $f_{\rm in,~ O}\sim 0.1$reproduces the observed trend between dust-to-gas ratio and metallicity, it is worth considering the effect of $\beta _{\rm SN}$ by setting $f_{\rm in,~ O}=0.1$.

Figure 1: Relation between dust-to-gas ratio and metallicity. The solid, dotted, and dashed lines show the model results with the dust destruction efficiencies (Eq. (2)) $\beta _{\rm SN}=1$, 5, and 25, respectively. The black and gray squares show the data for the blue compact dwarf galaxies in Sage et al. (1992) and in Lisenfeld & Ferrara (1998), respectively.

From Fig. 1, we see that the variance in dust-to-gas ratio is reproduced by an order-of-magnitude variation in $\beta _{\rm SN}$ (1-25 here). We note that LF98's value ( $\beta_{\rm SN}=5$) is within this range. Even if outflow does not efficiently occur in BCDs, we can explain the variance of the dust-to-gas ratio of BCDs with the various "destruction efficiency'' of dust, $\beta _{\rm SN}$, although we can never reject the importance of outflow in the framework of this paper. The important point is that we have demonstrated that dust destruction by SNe can play an important role in producing the variance of $\cal{D}$. The result also indicates that the difference in dust-to-gas ratio among the three lines becomes clear at an oxygen abundance, $12+\log ({\rm O/H})\sim 8$. Thus, we conclude

1.

that the dust destruction by SNe can vary the dust-to-gas ratio when the metallicity, $12+\log ({\rm O/H})$, reaches about 8 ($\sim$10% of the solar metallicity), and

2.

that the large variation of the dust-to-gas ratio among BCDs is explained by the variation of dust destruction efficiency, $\beta _{\rm SN}$.