H ii REGION LUMINOSITY FUNCTION OF THE INTERACTING GALAXY M51

The total number of M51 H ii regions listed in the original sample is about 19,600. The Hα luminosity of these H ii regions ranges from L = 10^35.5 erg s⁻¹ to 10^39.0 erg s⁻¹. This detection limit is similar to the deep survey of H ii regions in the dwarf galaxies of the Local Group (Kennicutt & Hodge 1980; Hodge et al. 1989a, 1989b; Hodge & Lee 1990). The total Hα luminosity of all these H ii regions is derived to be L = 10^41.2 erg s⁻¹. There are 3653 H ii regions with L > 10³⁷ erg s⁻¹, and their total luminosity corresponds to 70% of the total Hα luminosity of M51. There are only 160 H ii regions with L > 10³⁸ erg s⁻¹.

We derived the H ii LF for the original sample, as shown in Figure 3. The lower limit of Hα luminosities is determined by the observational threshold on the surface brightness. The H ii LF shows a maximum at L ≈ 10^36.1 erg s⁻¹ and declines rapidly as the luminosity decreases. This value is similar to the one for M33 H ii regions, L ≈ 10^35.9 erg s⁻¹, found from the completeness-corrected H ii LF by Wyder et al. (1997), and those for NGC 6822 and the Magellanic Clouds, L ≈ 10³⁶ erg s⁻¹ (Hodge et al. 1989a; Wilcots et al. 1991). Correcting this value for the adopted mean extinction value for M51 H ii regions leads to ≈10^37.1 erg s⁻¹. This value is similar to that for M42, ≈10^36.9 erg s⁻¹ (Scoville et al. 2001). Considering that the H ii LF for M33 shows a flattening for L = 10³⁵–10³⁶ erg s⁻¹ (Hodge et al. 1999), the rapid decline for L < 10³⁶ erg s⁻¹ in our H ii LF appears to be due to incompleteness in our catalog.

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The H ii LF appears to be fitted by a double power law with a break at L ≈ 10^37.1 erg s⁻¹. We used a double power-law function to fit the H ii LF:

$$\begin{eqnarray*} &&N(L){\it dL}\,{=}\,AL^{\alpha _{1}}{\it dL}\quad\alpha _{1}: {\rm Bright\; power\hbox{-}law\; index \quad for}\ L\,{\ge}\, {L}_{b}, \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&N(L){\it dL}\,{=}\,A^{\prime } L^{\alpha _{2}}{\it dL}\quad \alpha _{2}: {\rm Faint\; power\hbox{-}law\; index \quad for}\ L\,{<}\,{L}_{b}, \end{eqnarray*}$$

where L_b is the break point luminosity, and A' = AL^{(α₂ − α₁)}_b. For the bright H ii regions (10^37.1 erg s⁻¹ <L < 10^38.8 erg s⁻¹), we obtained a power-law index, α = −2.25 ± 0.02. For the faint part (10^36.1 erg s⁻¹ <L < 10^37.1 erg s⁻¹), the H ii LF becomes flatter than for the bright part, having α = −1.42 ± 0.01.

A distinguishable point with this H ii LF is a break at L = 10^37.1 erg s⁻¹. This break has not been observed in previous studies of the H ii regions in M51, because the data used in previous studies were not deep enough to detect it (Rand 1992; Petit et al. 1996; Thilker et al. 2000; Scoville et al. 2001). However, the H ii LF break at similar luminosity is known to exist for some nearby galaxies: MW (Paladini et al. 2009), M31 (Walterbos & Braun 1992) and M33 (Hodge et al. 1999). This break may be caused by the transition of H ii region ionizing sources, from low-mass clusters (including several OB stars) to more massive clusters (including several tens of OB stars; Kennicutt et al. 1989; McKee & Williams 1997; Oey & Clarke 1998). This will be discussed in detail in the next section.

We compared the H ii LF for the original sample (solid line) with that in Thilker et al. (2000; dashed line), as shown in Figure 4(a). The Hα luminosity of H ii regions for the original sample ranges from L = 10^35.5 erg s⁻¹ to 10^39.0 erg s⁻¹, while it ranges L = 10^36.4–10^39.4 erg s⁻¹ in Thilker et al. (2000). The minimum luminosity of the H ii regions in the original sample is about 10 times fainter than that in Thilker et al. (2000). While this study shows a break at L = 10^37.1 erg s⁻¹, the study by Thilker et al. (2000) shows a break at L = 10^38.8 erg s⁻¹, much brighter than that in this study. The power-law indices in this study are α = −2.28 ± 0.02 for the bright part and α = −1.38 ± 0.01 for the faint part, and those in Thilker et al. (2000) are α = −3.41 ± 0.08 and α = −1.61 ± 0.03.

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Figure 4(b) shows the H ii LF for the original sample (solid line) and that in Scoville et al. (2001, dashed line). The spatial coverage of Hα images used in this study is large enough to include most of the entire disk of NGC 5194 as well as its companion NGC 5195. However, Scoville et al. (2001) obtained the H ii LF from Hα images covering only the central 45 × 4' field of NGC 5194. For comparison, we plotted two H ii LFs derived for the same area. The minimum and maximum luminosities of the H ii regions in two samples are similar, but there is a difference in the numbers of H ii regions with L < 10^37.1 erg s⁻¹. The ACS images used in this study have longer exposure time than that of the WFPC2 images used in Scoville et al. (2001), so that we found many more faint H ii regions than Scoville et al. (2001). Both H ii LFs are fitted by a power law for the bright part (10^37.0 erg s⁻¹ <L < 10^38.6 erg s⁻¹). The H ii LF for the original sample is slightly steeper with α = −2.24 ± 0.03 than the H ii LF with α = −1.91 ± 0.04 reported by Scoville et al. (2001).

Figure 4(c) shows the H ii LF for the original sample (solid line) and that in Gutierrez et al. (2011, dashed line). The Hα luminosity of the H ii regions for the original sample ranges from L = 10^35.5 erg s⁻¹ to 10^39.0 erg s⁻¹, while it ranges L = 10^35.8 erg s⁻¹ to 10⁴⁰ erg s⁻¹ in Gutierrez et al. (2011). While this study shows a break at L = 10^37.1 erg s⁻¹, the study by Gutierrez et al. (2011) shows a break at L = 10^38.7 erg s⁻¹, much brighter than that in this study. The power-law indices in this study are α = −2.21 ± 0.02 for the bright part and α = −1.41 ± 0.02 for the faint part, and those in Gutierrez et al. (2011) are α = −2.29 ± 0.17 and α = −1.67 ± 0.03. This large discrepancy between the two studies is primarily due to two factors: (1) we considered the small local peaks for the large H ii regions in Gutierrez et al. (2011) as separate H ii regions in this study and (2) we found many isolated faint H ii regions, missed in Gutierrez et al. (2011).

In Figure 5(a) we compare the H ii LF for the group sample (solid line) with that for the original sample (dashed line). Although both H ii LFs are fitted by a double power law, the location of a break and the power-law indices are different. The locations of the break are L = 10^37.1 erg s⁻¹ and 10^38.8 erg s⁻¹, respectively. The power-law indices for the bright part are α = −2.21 ± 0.02 and −2.60 ± 0.21, and those for the faint part are α = −1.41 ± 0.02 and −1.69 ± 0.01 for the original and group samples, respectively.

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Figure 5(b) shows the H ii LFs for the grouped H ii regions (dotted line), the group sample (solid line), and the H ii regions in Thilker et al. (2000, dashed line). The locations of the break and the power-law indices for the latter two samples are similar. The locations of the break for both the group sample and Thilker et al. (2000) are L = 10^38.8 erg s⁻¹. The power-law indices for the group sample are α = −2.60 ± 0.21 for the bright part and α = −1.69 ± 0.01 for the faint part, and those in Thilker et al. (2000) are α = −3.41 ± 0.08 and α = −1.61 ± 0.03. In conclusion, the H ii LF break seen at L = 10^38.8 erg s⁻¹ in the previous studies based on the ground-based images (Rand 1992; Petit et al. 1996; Thilker et al. 2000) can be explained by the blending of multiple H ii regions.

In Figure 5(c) we compare the H ii LFs for the group sample (solid line), the grouped H ii regions (dotted line), and the H ii regions in Gutierrez et al. (2011) (dashed line). The H ii LFs for the latter two samples show nearly the same shape. The locations of the break and the power-law indices for the group sample and in Gutierrez et al. (2011) are similar. The location of the break for the group sample and that in Gutierrez et al. (2011) are L = 10^38.8 erg s⁻¹ and 10^38.7 erg s⁻¹, respectively. The power-law indices for the group sample are α = −2.60 ± 0.21 for the bright part and α = −1.69 ± 0.01 for the faint part, and those in Gutierrez et al. (2011) are α = −2.29 ± 0.17 and α = −1.67 ± 0.03. Both studies show a good agreement at L > 10^37.4 erg s⁻¹, however, there is a large difference in the number distribution of H ii regions at L < 10^37.4 erg s⁻¹. It is primarily due to the existence of many isolated faint H ii regions that were found in our survey, but missed in Gutierrez et al. (2011).

We investigated the spatial variation of the H ii LF for the original sample. We separated H ii regions in the original sample into four groups, according to their positions in the galaxy: arm, interarm, and nuclear regions of NGC 5194 and NGC 5195. The positions of the H ii regions are listed in Table 1. There are 12,245 H ii regions in the arm region, 4422 H ii regions in the nuclear region, and 2639 H ii regions in the interarm region of NGC 5194. We found only 292 H ii regions in the region of NGC 5195. The total luminosity of the arm H ii regions is derived to be L = 10^41.1 erg s⁻¹ and that of the nuclear H ii regions is L = 10^40.6 erg s⁻¹, while that of the interarm H ii regions is only L = 10^40.1 erg s⁻¹, 8% of the total Hα luminosity of all NGC 5194 H ii regions.

Figure 6(a) shows the H ii LFs for the arm (solid line), interarm (dashed line), and nuclear (dotted line) regions of NGC 5194. The minimum luminosities (L ≈ 10^35.5) in the H ii LFs are nearly the same in three regions, but the maximum luminosities are different depending on the position. The maximum luminosities for the arm and nuclear regions are L = 10^39.0 erg s⁻¹ and L = 10^38.8 erg s⁻¹, respectively. However, there is no H ii region brighter than L = 10^38.2 erg s⁻¹ for the interarm region.

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These H ii LFs are fitted by a double power law with a break at L = 10³⁷ erg s⁻¹. The power-law indices for the bright part are α = −1.92 ± 0.03, −2.01 ± 0.09, and −2.08 ± 0.05 for the arm, interarm, and nuclear regions, respectively. On the other hand, the power-law indices for the faint part are α = −1.33 ± 0.01, −1.53 ± 0.05, and −1.33 ± 0.03. Although the H ii LFs for the bright part (L > 10³⁷ erg s⁻¹) have similar power-law indices in three regions, the H ii LF for the faint part (L < 10³⁷ erg s⁻¹) is steeper in the interarm region than in the arm and nuclear regions.

Figures 6(b) and (c) shows the spatial variation of the H ii LFs for the grouped H ii regions and the group sample, respectively. The H ii LFs for the grouped H ii regions in Figure 6(b) are truncated at L = 10³⁷ erg s⁻¹. We used a single power law to fit the H ii LFs for the range from L = 10^37.0 erg s⁻¹ to L = 10^38.4 erg s⁻¹. The power-law indices are α = −1.57 ± 0.04, −1.74 ± 0.08, and −1.45 ± 0.06 for the arm, interarm, and nuclear regions, respectively. The H ii LF is steeper in the interarm region than in the arm and nuclear regions. On the other hand, the H ii LFs for the group sample in Figure 6(c) are truncated near L = 10³⁶ erg s⁻¹. The power-law indices are α = −1.78 ± 0.04, −1.95 ± 0.08, and −1.58 ± 0.06 for the arm, interarm, and nuclear regions, respectively. The H ii LF is also steeper in the interarm region than in the arm and nuclear regions.

The size distribution of the H ii regions in the original sample is displayed in Figure 7, showing that their sizes (diameters) range from 8 pc to 110 pc. Most of the previous H ii region studies show that the cumulative size distribution of the H ii regions in a galaxy can be well fitted with an exponential function (van den Bergh 1981; Hodge 1987; Cepa & Beckman 1990; Knapen 1998; Hakobyan et al. 2007). However, some other studies suggested that the power-law form fits the differential size distribution better than the exponential function (Kennicutt & Hodge 1980; Elmegreen & Salzer 1999; Pleuss et al. 2000; Oey et al. 2003; Buckalew & Kobulnicky 2006).

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Figures 7(a) and (b) display the cumulative and differential size distributions of the H ii regions, respectively. The cumulative size distribution of the H ii regions with 15 pc <D < 80 pc is fitted well with an exponential law: N(D) = N₀ exp (− D/D₀) where N₀ = 77, 880 and D₀ = 9.3 pc. The differential size distribution of these H ii regions is fitted by a double power law with a break at D = 30 pc. The power-law index for small H ii regions with 15 pc <D < 30 pc is α_D = −1.78 ± 0.04, whereas α_D = −5.04 ± 0.08 for large H ii region with 30 pc <D < 110 pc. The small H ii regions with D < 30 pc may be ionized by several OB stars. On the other hand, the large H ii regions with D > 30 pc are considered to be superpositions of small H ii regions, and they can be associated with stellar associations including several tens of OB stars.

Figure 8(a) shows the H ii LF for small H ii regions with D < 30 pc. Most of the small H ii regions with D < 30 pc are fainter than L = 10^37.1 erg s⁻¹, and only 1065 H ii regions are brighter than L = 10^37.1 erg s⁻¹ (solid line). Out of the H ii regions with D < 30 pc, we selected 4320 H ii regions which are neither blended with neighbor sources nor located in the crowded regions (dashed line). Most of them are in the range from L = 10^35.4 erg s⁻¹ to 10^37.2 erg s⁻¹. The maximum L for the isolated H ii regions corresponds to about 10 times of M42, with an extinction-corrected value, L ≈ 10^36.9 erg s⁻¹. On the other hand, Figure 8(b) shows the H ii LF for large H ii regions with D > 30 pc. There are 1808 H ii regions brighter than L = 10^37.1 erg s⁻¹and 632 H ii regions fainter than L = 10^37.1 erg s⁻¹.

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We compared the size distribution of the H ii regions in this study with those in Thilker et al. (2000), Scoville et al. (2001), and Gutierrez et al. (2011). Figure 9(a) plots the size distribution of the H ii regions in the original sample (solid line), the group sample (dashed line) and Scoville et al. (2001, dot-dashed line). In comparison with Scoville et al. (2001), we found that there are many more small H ii regions in the original sample. We used a single power law to fit the size distributions of H ii regions for the range from 30 pc to 125 pc. The power-law indices are α_D = −5.94 ± 0.26, −3.23 ± 0.10, and −2.79 ± 0.06 for the original sample, the group sample, and Scoville et al. (2001), respectively. Thus, the size distribution of the original sample is much steeper than that of the group sample and that of Scoville et al. (2001) sample, while the latter two samples have similar power-law indices.

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Figure 9(b) shows the size distribution of the grouped H ii regions (solid line), the H ii regions in the group sample (dashed line), in Thilker et al. (2000, dotted line) and in Gutierrez et al. (2011, dot-dashed line). The sizes of the H ii regions in the group sample range from 8 pc to 310 pc, while those of the H ii regions in Thilker et al. (2000) range from 80 pc to 480 pc. This large discrepancy between the two studies is considered to come from the different spatial resolutions. We found that there are many more small H ii regions in the group sample, compared with Gutierrez et al. (2011). In comparison between the grouped H ii regions and the H ii regions in Gutierrez et al. (2011), we found that although the size distribution for the small H ii regions (D < 50 pc) have similar shapes, the size distribution for the large H ii regions (D > 50 pc) shows different shapes, having a steeper power slope for the grouped H ii regions. The power-law indices are α_D = −3.48 ± 0.08 and −2.75 ± 0.11 for the grouped H ii regions and in Gutierrez et al. (2011), respectively. In other words, the size of the H ii regions derived from this study is generally smaller than in Gutierrez et al. (2011). It is because we derived the sizes of the grouped H ii regions from summing the numbers of pixels contained in all individual H ii regions in a group, while Gutierrez et al. (2011) considered the substantial empty volumes to measure the size of H ii regions.

In order to understand the nature of the observational H ii LF, we performed Monte Carlo simulations of H ii LFs, following Oey & Clarke (1998). The Hα luminosity of an H ii region depends on (1) the stellar initial mass function (IMF) and (2) the relation of Lyman continuum emission rate (Q_Lyc) and stellar mass. We adopted a Salpeter IMF (Salpeter 1955) with N(m_*)dm_*∝m_*^−2.35 over the range from 1 M_☉ to 100 M_☉. We used Q_Lyc, depending on stellar mass, given by Vacca et al. (1996) which is based on stellar models with constraints provided by observations of individual high-mass stars.

We considered artificial clusters with the number of stars per cluster (N_*) ranging from 1 to 10,000 and estimated the total Q_Lyc radiating from each cluster. To reduce the statistical noise, we repeated the experiment 10,000 times, and derived the average Q_Lyc of the individual cluster with a specific N_*. Then we obtained the Hα luminosity of this cluster using L = 1.37 × 10⁻¹²Q_Lyc given in Scoville et al. (2001).

We represented the distribution of numbers of stars in each model cluster by a power law: N(N_*)dN_* = N_*^βdN_*, where N(N_*)dN_* is the number of stars per cluster in the range from N_* to N_* + dN_* and β is a power-law index. We assumed that the star formation in each cluster occurs on a short timescale compared with stellar evolution timescales and that all clusters in a galaxy are created at the same time. We converted simulated H ii LFs into observational H ii LFs for comparison, considering the internal extinction as adopted above.

The shape of the H ii LFs is determined by two parameters: β and the maximum number of stars per cluster, N_{*, max}. To know how these two parameters affect the H ii LF, we constructed nine H ii LFs with 300,000 zero-age clusters for three power-law indices (β = −1.60, −2.00, and −2.40) and three values of N_{*, max} (330, 1000, and 10,000), plotting them in Figure 10. The masses of clusters with N_{*, max} = 330 and β = 2.0 are 1000 M_☉ and 2500 M_☉, respectively, for the stellar mass range of 1–100 M_☉ and 0.1–100 M_☉. The masses of clusters with N_{*, max} = 1000 are respectively 3090 M_☉, and 7700 M_☉, and the masses of clusters with N_{*, max} = 10, 000 are respectively 30,900 M_☉ and 77,000 M_☉, for the same ranges of stellar mass. The number of OB stars with M > 17 M_☉ are ∼7, ∼20, and ∼60 for N_{*, max} = 330, 1,000 and 10,000, respectively.

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The H ii LFs with N_{*, max} = 330 in the upper panels of Figure 10 are truncated at L = 10^37.0 erg s⁻¹. Clusters with N_* < 330 produce few H ii regions brighter than L = 10^37.0 erg s⁻¹. They show nearly flat slopes for β = −1.60 and −2.00, but show some slope for β = −2.40. In the middle panels, the H ii LFs with N_{*, max} = 1000 show the existence of some bright H ii regions with L > 10^37.0 erg s⁻¹. They show a change in slope at L ≈ 10^37.0 erg s⁻¹, which is similar to the value found for M51 H ii regions in this study (Figure 3). We fit the bright part (L > 10^37.0 erg s⁻¹) with a power law, obtaining α = −1.48, −1.86, and −2.23 for β = −1.60, −2.00, and −2.40, respectively. Thus the values of α decreases as β decreases. The H ii LFs with N_{*, max} = 10,000 in the low panels show a more obvious break at L = 10^37.0 erg s⁻¹. Fitting the bright part (L > 10^37.0 erg s⁻¹) yields α = −1.55, −1.94, and −2.36 for β = −1.60, −2.00, and −2.40, respectively. Thus the values of α are very similar to those for β, showing that the values of these two parameters get similar as N_* increases.

For comparison of the H ii LFs in Figure 10 with the observational ones, we need to consider the effect of evolution of Q_Lyc. We assumed that Q_Lyc remains constant during the main-sequence lifetimes and is zero thereafter. Main-sequence lifetimes (t_ms) for individual OB stars are obtained from Maeder (1987). Figure 11 shows the H ii LFs with β = −2.40 and N_{*, max} = 10,000 derived for four ages 0, 2, 4, and 6 Myr. We adopted a value for β, similar to the observational value derived in this study. The number of bright H ii regions decreases as ages increase. Fitting the bright part (L > 10^37.0 erg s⁻¹) yields α = −2.35, −2.39, and −2.35 for age = 0, 2, and 4 Myr, respectively. There are few bright H ii regions in the case of age = 6 Myr, but a fit including the faint part (L > 10^36.0 erg s⁻¹) yields a similar index, α = −2.35. Thus the values of α change little depending on age, and they are almost the same as the input value of β.

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A controversial point in the H ii LF for M51 has been a break at L ≈ 10^38.8 erg s⁻¹ shown in previous studies based on the ground-based imaging (Kennicutt et al. 1989; Rand 1992; Thilker et al. 2000). Kennicutt et al. (1989) argued that the cause of this break might be a transition from the normal to the super giant H ii regions. On the other hand, the break is possibly a feature caused by the low spatial resolution. Scoville et al. (2001) compared their H ii LF of M51 derived from HST WFPC2 with that of Rand (1992) based on the ground-based images. They demonstrated that the high-resolution H ii LF is significantly steeper than the low-resolution one and showed that the break at L ≈ 10^38.8 erg s⁻¹ is not seen in their H ii LF.

This break is not seen either in the H ii LF for the original sample in this study, as shown in Figure 3. However, we can observe a break at L = 10^38.8 erg s⁻¹ in the H ii LF for the group sample in Figure 5. From the similarity between this H ii LF and that in Thilker et al. (2000) in Figure 5(b), we conclude that the break seen at L ≈ 10^38.8 erg s⁻¹ in the previous studies may be due to the blending of multiple H ii regions.

On the other hand, we found a break at L = 10^37.1 erg s⁻¹ in the H ii LF for the original sample as shown in Figure 3. To understand the shape of the H ii LF with this break we compared the observational H ii LF for the original sample with the simulated H ii LF in Figure 10. The shape of the observational H ii LF is remarkably similar to that of the simulated H ii LF in Figure 10(i) derived for β = −2.40 and N_{*, max} = 10,000. The observational H ii LF shows a break at L = 10^37.1 erg s⁻¹, which is consistent with the simulated H ii LF with a break at L = 10^37.0 erg s⁻¹. The power-law indices of the observational H ii LF are α = −2.25 ± 0.02 for the bright part and α = −1.42 ± 0.01 for the faint part, agreeing with those in the simulated H ii LF, α = −2.36 ± 0.03 and α = −1.43 ± 0.02. Thus, it is concluded that the break at L = 10^37.1 erg s⁻¹ is due to the different N_* in clusters associated with individual H ii regions. In other words, the faint H ii regions with L < 10^37.1 erg s⁻¹ are ionized by low-mass clusters with N_* < 330 (including several OB stars) or single massive stars, while the bright H ii regions with L > 10^37.1 erg s⁻¹ are ionized by massive clusters with N_* ≫ 330 (including several tens of OB stars).

It is noted that the cluster mass M_cl is related with N_*: M_cl∝N_*. Therefore, the power-law index α of H ii LF in the bright end is expected to be similar to the index β (–2.25) of the IMF of the clusters in M51 that may be represented by a power law: N(M_cl)dM_cl = M_cl^−2.25dM_cl. This slope (β) is similar to or slightly steeper than those derived from previous studies for M51 young clusters: from the observation of young star clusters in M51, the cluster mass function is known to have power-law form with its index ranging from α = −1.70 ± 0.08 (Gieles et al. 2006) to α = −2.23 ± 0.34 (Hwang & Lee 2010).

Figure 6 shows that the shape of the H ii LFs varies depending on the location within a galaxy. In Figure 6(a) for the original sample, we found that although the H ii LFs for the bright part (L > 10³⁷ erg s⁻¹) have similar power-law indices in three regions, the H ii LF for the faint part (L < 10³⁷ erg s⁻¹) is steeper in the interarm region than in the arm and nuclear regions. In addition, there is no H ii region brighter than L = 10^38.2 erg s⁻¹ in the interarm region. In Figures 6(b) and (c) for the grouped H ii regions and the group sample, we also found that the H ii LF is steeper in the interarm region than in the arm and nuclear regions.

Although the shape of the H ii LF can be affected by several factors, including variation in the metallicity, extinction, stellar IMF, and underlying gas dynamics, Kennicutt et al. (1989) argued that the most significant factor is the change in the total number of stars produced in typical star formation events. Rand (1992) attributed the difference between the arm and the interarm H ii LF to the existence of a more massive molecular clouds in the arm region. In other words, the higher proportion of massive clouds gives rise to relatively more H ii regions of the higher luminosity. It is reasonable to infer that the high gas surface densities combined with the compression by the strong density wave in the arms may cause a concentration of bright H ii regions on the arm region. Later Oey & Clarke (1998) suggested that the interarm H ii regions are on average fainter than those in the arm region because the ionizing sources in the arm region are younger than those in the interarm region.

In Figure 11, we show the evolution of H ii LFs with β = −2.40 and N_{*, max} = 10,000 for different ages (0, 2, 4, and 6 Myr). The H ii LFs shift to faintward as a function of time. A maximum luminosity after 4 Myr (panel (c)) is 0.5 dex fainter than at zero-age (panel (a)), and the power-law index for the faint part (L < 10^37.1 erg s⁻¹) after 4 Myr is steeper than at zero age. The interarm H ii LF in Figure 6 has a lower maximum luminosity and a steeper slope for the faint part than the arm H ii LF. Thus the difference between the arm and the interarm H ii LFs can be explained by an effect of evolution, supporting the suggestion by Oey & Clarke (1998).

Both the Hα luminosity function and the size distribution of H ii regions in the original sample are fitted by a double power law as shown in Figures 3 and 7(b), respectively. Since the total Hα luminosity of an H ii region is integrated from the nebular volume emission, it is expected that L∝D³. Then the power-law indices (α and α_D) of the Hα luminosity function and the size distribution for H ii regions are related as α_D = (2–3)α (Oey et al. 2003).

The H ii LF for the original sample in Figure 3 shows that the break appears at L = 10^37.1 erg s⁻¹, and that the power-law indices are α = −1.42 ± 0.01 for the faint part and α = −2.25 ± 0.02 for the bright part. Using these values we derived expected values, α_D = −2.26 and α_D = −4.75 for the faint and bright part, respectively. On the other hand, the observational size distribution in Figure 7(b) shows that the power-law index for the small H ii regions with D < 30 pc is α_D = −1.78 ± 0.04, whereas α_D = −5.04 ± 0.10 for the large H ii regions with D > 30 pc. Therefore, there is a good agreement between expected and observational indices for the size distribution of H ii regions.

Figure 12(a) displays Hα luminosities versus sizes of the H ii regions in the original sample. It shows a clear correlation between the two parameters. The relation between the luminosity and the size of H ii regions is fitted by L∝D^{3.04 ± 0.01} (solid line). This value is in good agreement with our expectation of L∝D³. However, Scoville et al. (2001) derived a flatter relation, L∝D^{2.16 ± 0.02} for H ii regions in the central region of M51 (dashed line). They explained that the this flatter slope is due to blending effect: large H ii regions are blended so that they include substantial empty or neutral volumes. This indicates that this study separated blended H ii regions better than Scoville et al. (2001).

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Figures 12(b) and (c) display Hα luminosities versus sizes of the grouped H ii regions and the H ii regions in the group sample, respectively. The relation between the luminosity and the size of H ii regions for the two samples is fitted by L∝D^{2.78 ± 0.02} (panel (b); solid line) and L∝D^{2.78 ± 0.01}(panel (c); solid line). Gutierrez et al. (2011) derived a flatter relation, L∝D^{2.29 ± 0.03} (dashed line). This discrepancy between the two studies is due to the different procedure used to derive the size of the H ii region, as mentioned in Section 5.3.