Proceedings of the Workshop
"The Magellanic Clouds and Other Dwarf Galaxies"
of the Bonn/Bochum-Graduiertenkolleg

Interpreting the H II Region Luminosity Function

M.S. Oey and C.J. Clarke

Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK

Received 06th March 1998
Abstract. We construct Monte Carlo simulations of the H II region luminosity function (H II LF), drawing ionizing stars from a constant stellar IMF, and the number of ionizing stars from a power-law distribution of constant slope. We find that observed variations in the form of the H II LF across the Hubble sequence can be explained by a trend in the maximum number of ionizing stars per nebula. In addition, variations in the form of the H II LF between arm and interarm populations of spiral galaxies can be explained by evolutionary effects. The H II LF can thus reveal features in the most recent (< 10 Myr) star formation history of the host galaxies.

1. Introduction

The H II region luminosity function (H II LF) is an important probe of current star formation properties in nearby galaxies. What can the form of the H II LF reveal about the underlying stellar ionizing populations? This contribution summarizes our investigation of this question (Oey & Clarke 1998), using Monte Carlo simulations of the stellar populations.

The observed, differential H II LF is usually represented by a power-law of the form N (L) \ dL ∝ L-a \ dL, where N (L) \ dL is the number of objects with luminosities in the range L to L + dL. Observations of the H II LF in nearby galaxies reveal three intriguing patterns:

  1. A break in slope is often reported (e.g., Kennicutt et al. 1989, hereafter KEH; Walterbos & Braun 1992; Rozas et al. 1996), such that lower-L populations show a flatter slope than the brighter populations;
  2. The H II LF s of arm populations in grand-design spirals sometimes show a shallower slope than those in interarm regions (e.g., KEH; Banfi et al. 1993);
  3. There is a correlation with galactic Hubble type, such that galaxies of the latest types, Sc - Im, show H II LF slopes of a ∼ 1.7; Sb types show a ∼ 2.0 (KEH; Banfi et al. 1993); and Sa galaxies show a much steeper a ∼ 2.6 (Caldwell et al. 1991).

2. Monte Carlo models

To understand these variations, we modeled the H II LF using a Monte Carlo technique to draw the stellar ionizing populations. We assumed a constant, Salpeter (1955) initial mass function (IMF), with an upper-mass limit mup = 100 Msun, and considered ionizing stars with masses m down to a lower-mass limit mlo = 17 Msun. We used simple power-laws to relate the stellar mass to its contributed Halpha luminosity l and main-sequence lifetime tms:

mlδ   and   mtms-d.         (1)

Based on the stellar models of Schaerer et al. (1993), we adopt values of δ = 1.5 and d = 0.7 for our stellar mass range. Below our adopted mlo, these power-law relations steepen significantly, hence the H II LF is strongly dominated by the contributions of the higher-mass stars.

The H II LF is closely related to the underlying mass function of the ionizing clusters. Hence, we also use a power-law to describe the distribution of N*, the number of stars per cluster: N (N*) \ dN*N* \ dN*. To model the H II LF, we randomly drew N* from a fixed power-law distribution of slope β = 2. We then randomly drew the N* stellar masses for each cluster from the IMF described above, and used the stellar m - l relation (Eq. 1) to compute the nebular luminosity L.

[Click here to see Fig. 1!]

The luminosity evolution of the nebulae, in conjunction with the creation history of the objects, significantly affects the form of the H II LF. We consider two creation histories for the nebular population: a single burst, and a constant creation rate. We also consider two models for the luminosity evolution of the stellar ionizing fluxes. In the first case, we assume that l remains constant for the duration of tms and is zero thereafter; in the second case, we assume a delayed power-law fading of l. The two models for l(t) produce H II LF s that do not differ dramatically, qualitatively. We therefore present only results for the first model of l(t) here. A complete presentation of the second model may be found in Oey & Clarke (1998).

2.1. Single burst models: arm and interarm regions

In Fig. 1, we show our simulations of the H II LF for the single burst creation scenario. Figure 1a shows the zero-age model, and Fig. 1b shows the model at an age of 7 Myr. The actual age representation should not be taken too literally, owing to the crude parameterizations of the stellar properties (Eq. 1).

The most important feature in the zero-age model is the two-slope structure in the H II LF. This behavior was first found by McKee & Williams (1997), and is caused by the small-number statistics of the stellar population in the low-L objects. In the high-L tail of the H II LF, the stars are fully populating the IMF in the ionizing clusters. Hence, the mean L per cluster is constant, and therefore the nebular LN*, and a = β. We term this population of objects as ``saturated'' with respect to the IMF. On the other hand, for the low-L H II regions, a given nebular luminosity L can result from a variety of stellar combinations, for example, a larger number of low-mass stars, or a smaller number of higher-mass stars. The mean L per cluster therefore varies, and L is not related to N* in a simple analytic fashion. We term these objects as ``unsaturated'' with respect to the IMF. The H II LF slope for the unsaturated population is flatter than that for the saturated, because of the larger range of L contributed by a fixed bin dN* for the unsaturated objects. The location of the turnover in the H II LF generally, but not always, occurs at lup, the Halpha luminosity contributed by mup. We caution that in our models, the slope for the unsaturated objects is flatter than would be expected in reality, because of the power-law approximation for the m-l relation (Eq. 1). Further details on these issues are given by Oey & Clarke (1998).

[Click here to see Fig. 2!]

Figure 1 shows that for the single burst case, the primary effect of nebular luminosity evolution is simply to shift the entire H II LF to lower L as the population ages, at a rate tδ/d. Note that in the evolved model, the saturated objects do extend below lup, and are augmented by nebulae that were originally unsaturated in the highest-L stars, but later qualify as saturated, as the highest-mass stars in the entire population expire. These qualitative behaviors are also true for the stellar luminosity evolution described by the delayed power-law fading function for l(t), in which case the H II LF evolves even faster in luminosity. The latter model for l(t) also produces a noticeable, sharp peak near the transition between saturated and unsaturated populations in the evolved models (see Oey & Clarke 1998).

We suggest that this evolutionary behavior in the H II LF is observed in the variation between arm and interarm populations of nebulae in grand-design spiral galaxies. This is exactly the expected behavior if the arm H II regions represent the current burst of star formation, leaving the interarm regions with significant populations of aging nebulae remaining in the wake of the arms. There are published examples of H II LFs for arm and interarm regions in six spiral galaxies: NGC 157, NGC 3631, NGC 6951 (Rozas et al. 1996); NGC 6814 (Knapen et al. 1993); M51 (Rand 1992); and M100 (Knapen 1997). In each case, the interarm population has a maximum L that is less than that for the arm population, and additionally, the H II LF peak in each case is also at a lower L for the interarm regions. Thus the entire interarm H II LF appears shifted to lower luminosities from the arm H II LF. This effect will also yield a flatter measured slope for the arm populations over a fixed range of L, since it would actually represent a composite slope of both the unsaturated and saturated populations; whereas the interarm population would represent primarily the steeper, saturated slope alone. Thus, this could also explain why flatter slopes are sometimes reported for the arm regions.

2.2. Continuous creation models: the Hubble Sequence

Figure 2 shows our simulations for constant, continuous creation of the H II regions. The constant creation scenario can be considered as a superposition of many individual bursts. Hence, the slope of the saturated tail will remain unaffected, since it always reflects β at L > lup. The slope at L < lup, however, will steepen to a value intermediate between its value in the flatter, zero-age model, and β, as the saturated objects of differing ages contribute to the H II LF at those luminosities. This behavior can be seen in the comparison of Fig. 2a with Fig. 1a.

Figure 2b is the same as Fig. 2a, but showing only unsaturated objects. We use the same power-law in N*, having slope β = 2, but truncate the distribution to use only N* ≤ 10. For our stellar mass range, the transition N* between unsaturated and saturated clusters is between 10 and 20 stars. It is apparent that the H II LF is the same between Figs. 2a and 2b in the regime dominated by unsaturated objects, but that the high-L tail now shows an extremely steep dropoff in Fig. 2b where the saturated population is missing.

We suggest that the observed changes in the H II LF with Hubble sequence can be explained by a progression in maximum N*, such that the latest-type galaxies have no apparent maximum N*, and the earliest disk-type galaxies have a cutoff yielding only unsaturated objects. The extremely steep slopes of a ∼ 2.6 reported for Sa galaxies (Caldwell et al. 1991) are likely to reflect the steep dropoff seen in Fig. 2b, caused by the lack of saturated objects. On the other hand, observed slopes of a ∼ 1.7 or 2.0 for late-type galaxies (KEH) are likely to reflect β. The maximum L seen in the Sa H II LF s is around 1038 erg s-1lup, consistent with a fully unsaturated population; whereas the maximum L is progressively higher in later galactic types. It may be possible that the latest-type galaxies, including dwarf irregulars, show composite slopes of saturated and unsaturated populations of nebulae, as suggested above for some measurements of arm regions in spiral galaxies. Likewise, slope measurements for intermediate Hubble types are probably composite slopes of β and the upper end dropoff. Indeed, a trend in upper-L cutoff of the H II LF as a function of Hubble type has already been suggested by KEH and McKee & Williams (1997).

Thus, we demonstrate that the measured changes in the H II LF slope a do not necessarily reflect changes in the underlying slope β of the ionizing cluster mass function. It will be interesting to reexamine the H II LF s as a function of Hubble type in light of our models, to see whether a universal β is warranted. For example, Elmegreen & Efremov (1997) find a constant slope of β = 2 for both globular and open clusters, in both the Galaxy and Magellanic Clouds.

3. Conclusion

Our simulations show that the form of the H II LF can reveal important patterns in the nature and history of star formation. The differences seen in arm vs. interarm populations in spiral galaxies can be interpreted as an evolutionary effect. The changes across the Hubble sequence can be caused by a change in maximum N* for a universal, β ≅ 2 power-law in the ionizing cluster mass function. Hence, the form of the H II LF can distinguish between an evolved single burst, but full distribution in N*, and a continuous creation of unsaturated objects. There are two mechanisms in our models that can cause a break in slope: if the break occurs at Llup, it is likely to be caused by the transition between unsaturated and saturated objects; whereas if the break occurs at higher L, it can be explained by the dropoff caused by an upper cutoff in the N* distribution.
Acknowledgments. We are grateful for discussions with J. Beckman, B. Elmegreen, C. Feinstein, R. Kennicutt, and J. Knapen.

References


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First version: 20thMarch,1998
Last update: 14thNovember,1998

Jochen M. Braun   &   Tom Richtler
 (E-Mail: jbraun|richtler@astro.uni-bonn.de)