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:
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*-β \ 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.
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).
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 L ∝ N*, 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).
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.
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-1 ∼ lup, 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.