On the Form of the HII Region Luminosity Function

Abstract

Observed variations in the HII region luminosity function (HII LF) seen in spiral arm vs. interarm regions, and different galactic Hubble type, can be explained simply by evolutionary effects and maximum number of ionizing stars per cluster. We present Monte Carlo simulations of the HII LF, drawing the number of ionizing stars N_* from a power-law distribution of constant slope, and the stellar masses from a Salpeter IMF with an upper-mass limit of 100 M_sol. We investigate the evolution of the HII LF, as determined by stellar main-sequence lifetimes and ionizing luminosities, for a single burst case and continuous creation of the nebular population. Shallower HII LF slopes measured for the arms of spiral galaxies can be explained as a composite slope, expected for a zero-age burst population, whereas the interarm regions tend to be dominated by evolved rich clusters described by a single, steeper slope. Steeper slopes in earlier-type galaxies can be explained simply by a lower maximum N_* cutoff found for the parent OB associations. The form of the HII LF can reveal features of the most recent (~< 10 Myr) star formation history in nearby galaxies.

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arXiv:astro-ph/9712157v1 11 Dec 1997
On the form of the HII region luminosity function
Accepted 10-Dec-97 to the Astronomical Journal
M. S. Oey and C. J. Clarke
Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, U.K.
oey@ast.cam.ac.uk; cclarke@ast.cam.ac.uk
ABSTRACT
Observed variations in the H II region luminosity function (H II LF) seen in spiral
arm vs. interarm regions, and different galactic Hubble type, can be explained simply
by evolutionary effects and maximum number of ionizing stars per cluster. We present
Monte Carlo simulations of the H II LF, drawing the number of ionizing stars N∗from
a power-law distribution of constant slope, and the stellar masses from a Salpeter IMF
with an upper-mass limit of 100M⊙. We investigate the evolution of the H II LF, as
determined by stellar main-sequence lifetimes and ionizing luminosities, for a single
burst case and continuous creation of the nebular population. Shallower H II LF slopes
measured for the arms of spiral galaxies can be explained as a composite slope, expected
for a zero-age burst population, whereas the interarm regions tend to be dominated by
evolved rich clusters described by a single, steeper slope. Steeper slopes in earlier-type
galaxies can be explained simply by a lower maximum N∗cutoff found for the parent
OB associations. The form of the H II LF can reveal features of the most recent (∼< 10
Myr) star formation history in nearby galaxies.
Subject headings: H II regions — galaxies: fundamental parameters — galaxies: ISM —
galaxies: star clusters — ISM: structure — stars: formation
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1.Introduction
H II regions have long served as a primary indicator
of high-mass star formation, both in the Milky Way
and in external galaxies. The luminosity function of
H II regions (HII LF) is therefore a vital probe of the
present global star formation properties in individual
galaxies. In recent years, the number and accuracy of
compiled HII LFs for nearby galaxies has increased
substantially, extending the use of this tool to greater
distances.
The differential HII LF is usually parameterized as
a power law:
N(L) dL = A L−adL,(1)
where N(L) dL is the number of nebulae with lumi-
nosities in the range L to L + dL. Some interesting
patterns have emerged regarding the form of the HII
LF:
1. A large number of galaxies show a break in
slope of the HII LF with the fainter H II re-
gions showing a shallower slope compared to the
high-luminosity objects (e.g., Kennicutt, Edgar,
& Hodge 1989, hereafter KEH; Rand 1992;
Walterbos & Braun 1992; Rozas, Beckman, &
Knapen 1996a;)
2. The HII LF for arm and interarm regions of
spiral galaxies sometimes show steeper slopes in
the interarm regions(e.g., KEH; Banfi etal. 1993;
Rand 1992). However, there are also many in-
stances where no difference is found (e.g., Rozas
etal. 1996a; Knapen etal. 1993; Knapen 1997)
3. The slope of the HII LF is correlated with galac-
tic Hubble type, such that early-type galaxies
show steeper slopes than late-types. The power-
law index a ∼ 2.0 for Sb – Sc galaxies, compared
to a ∼ 1.7 for Sc – Im galaxies (KEH; Banfi
etal. 1993). It is even steeper in Sa galaxies,
with a ∼ 2.6 (Caldwell etal. 1991).
These differences in the HII LF have usually been
interpreted as resulting from corresponding differ-
ences in star formation properties. For example, the
break in slope (Point 1, above) has been suggested
to be caused by a physical transition between normal
H II regions and the class of supergiant H II regions
such as 30 Doradus in the Large Magellanic Cloud
(KEH). Likewise, the difference in slope for arm vs.
interarm regions, and among various Hubble types
(Points 2 and 3), have been attributed to differences
in gasdynamics and molecular cloud mass spectrum
(e.g., Thronson, Rubin, and Ksir 1991; Rand 1992;
KEH). However, we feel that it may be premature to
draw a direct connection between the form of the HII
LF and details of star forming environment such as
these. There are other effects that can influence the
form of the HII LF, that have not been adequately
explored.
One important effect, in particular, is the influence
of simple evolution in the ionizing clusters, and hence,
luminosity of the host H II regions. There are several
empirical factors that suggest that nebular luminosity
evolution is an important effect in the form of the HII
LF. The steeper slopes seen in interarm regions of spi-
ral galaxies (Point 2, above) have been identified by
von Hippel & Bothun (1990) as evidence of an aging
effect in those populations. Assuming star formation
takes place primarily in the spiral density waves de-
lineated by the arm regions, then the interarm regions
should exhibit a typically older population of nebulae
in the wake of this star formation activity. In addi-
tion, von Hippel & Bothun cite the behavior of Hα
equivalent widths as further compelling evidence of
the importance of luminosity evolution. As a clus-
ter ages, its associated Hα emission decreases, while
that of the underlying red stellar continuum increases,
thereby reducing the Hα equivalent width as a func-
tion of time. They demonstrate that for the H II re-
gions in NGC 628, there is indeed a clear decrease in
Hα equivalent width with decreasing Hα luminosity,
indicating higher numbers of evolved nebulae among
the fainter populations. Finally, Knapen etal. (1993)
and Rozas, Knapen, & Beckman (1996b) show that
the slope of the nebular luminosity vs. volume rela-
tion is slightly flatter in the interarm regions of five
galaxies as compared to the arm regions. As empha-
sized by the authors, this may be caused by obser-
vational selection effects, but if real, we suggest that
it is consistent with a higher tendency for shell-like
geometries in the interarm nebulae, as would be ex-
pected for an older population.
All of these arguments suggest that evolved neb-
ulae will be an important population in the HII LF.
Von Hippel & Bothun (1990) carried out the first de-
tailed investigation of the evolutionary effect on the
HII LF. They constructed two models to reproduce
the observed slope for NGC 628: one with zero evolu-
tion, in which the power-law slope of the luminosity
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function is due entirely to the cluster mass spectrum;
and one in which the distribution in nebular luminos-
ity is due entirely to a different age and number dis-
tribution of objects with the same initial cluster mass.
The HII LF resulting from the evolutionary model is
sensitive to the stellar initial mass function (IMF),
since the relative numbers of stars at a given ion-
izing luminosity become important when these stars
become the dominant contributor to the nebular lu-
minosity. Recognizing this, von Hippel & Bothun ori-
ented their study toward exploring the use of the HII
LF to place constraints on the IMF. As a result, they
did not explicitly examine the consequences of ag-
ing on the form of the HII LF itself. In addition,
their assumption of uniform initial cluster masses in
the evolutionary model is of only limited use, since
actual initial cluster masses apparently have a power-
law distribution (e.g., Oey & Clarke 1997; see below).
Therefore, we will here investigate the effect of cluster
aging on the form of the HII LF itself.
2.Saturated Population
We have carried out a preliminary investigation
of this problem as part of a different study (Oey &
Clarke 1997), and review the important relevant is-
sues here. A first critical point is the fact that not all
clusters have the same relative contribution to ioniz-
ing luminosity from the various stellar masses. Such a
situation is true for rich clusters that sample the IMF
well, but breaks down for sparse clusters in which
small-number statistics determine the relative ioniz-
ing contribution of different stellar masses. We shall
refer to the rich clusters with good stellar statistics as
“saturated” with respect to the IMF, and those with
poor statistics as “unsaturated.” The importance of
this fundamental difference in the dominant ioniz-
ing contributors between the saturated and unsatu-
rated clusters is demonstrated by a break in the pre-
dicted slope of the unevolved HII LF. This change in
slope was clearly demonstrated by McKee & Williams
(1997; hereafter MW97), who performed Monte Carlo
simulations of clusters drawn from a truncated power-
law distribution in the numbers of member stars N∗,
and a Scalo (1986) IMF. Since L ∝ N∗for the satu-
rated objects, the HII LF slope is the same as the par-
ent slope in N∗; but for unsaturated populations, the
HII LF slope is distinctly flatter. This flattening is
caused by the increased scatter in the L’s contributed
by a bin of given N∗. The unsaturated condition oc-
curs when this scatter is large compared to the mean
L at the given N∗, hence the slope of the HII LF
flattens. MW97 observed that in their simulations,
this change in slope occurs at Lsat = lup, the lumi-
nosity contributed by a single star at the upper-mass
limit mup, of the IMF. Stellar models currently sug-
gest that lup is in the range 38.0∼< loglup∼< 38.5
(Schaerer & de Koter 1997; Vacca, Garmany, & Shull
1996; Panagia 1973). We caution however, that the
slope break Lsat in the HII LF does not necessarily
occur at lup, as we show in §4.3.
Investigation of evolutionary effects on the HII LF
should therefore consider the differences in stellar ion-
izing populations between the saturated and unsatu-
rated regime. We examined the behavior of the sat-
urated case analytically in Oey & Clarke (1997). We
considered two extremes in the formation of ioniz-
ing clusters: a single-burst scenario, in which all ob-
jects are created at the same time; and a continuous-
creation scenario. In what follows, we will also assume
a constant IMF throughout.
For the case of the single burst, it is apparent that,
since all objects follow the same luminosity evolution,
the initial slope of the HII LF is simply preserved.
However, in the case of continuous creation, such a
conclusion is not obvious. We described the luminos-
ity evolution of the H II regions in terms of a fading
function f(t) (Oey & Clarke 1997):
L = L0f(t), (2)
where L0is the zero-age nebular luminosity. Stellar
population synthesis work (e.g., Leitherer & Heckman
1995; Beltrametti, Tenorio-Tagle, & Yorke 1982) sug-
gests that the tail of the fading function f(t) is well-
described by a power-law in time t, so that
f =





1 , t < tms
?
t/tms(mup)
?−η
,t ≥ tms(mup)
(3)
For saturated objects, the onset time, tms(mup), for
the fading is the same for all objects. Nebular fading
causes initially bright objects to contribute to lower-L
bins at later ages, thus if the epoch of fading domi-
nates over the epoch of zero-age luminosity, we would
expect the observed HII LF to be steeper than the
unevolved one. We found that for f(t) of the form in
equation 3, and reasonable ranges for mup, tms(mup),
and η, that essentially no change in HII LF slope is
expected. There is an exception for initially shallow
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Fig. 1.— Evolution of ionizing photon emission rate q0 (sec−1) from Schaerer & de Koter (1997). The stellar
models are, from bottom, 20, 25, 40, 60, 85, and 120M⊙.
slopes of < η−1+ 1, for which the observed slope in-
deed steepens to a value a = η−1+ 1. This critical
value of the slope is probably in the range 1.2 – 1.4,
which is lower than most measured HII LF slopes,
hence it is likely that this effect is unimportant. These
effects are described in further detail in Oey & Clarke
(1997). Thus in the saturated regime, no change in
HII LF slope is reasonably expected.
3. Monte Carlo Models
The statistical uncertainties that dominate the un-
saturated clusters render an analytic investigation of
these objects unfeasible.
the fact that many different combinations of stellar
masses can contribute to any given luminosity, and
there is no simple way to relate the total nebular lu-
minosity to the stellar composition, and hence, lumi-
nosity evolution. We have therefore modeled the HII
LF with Monte Carlo simulations to investigate its
behavior in the unsaturated regime.
Our model draws the integer number of ionizing
stars from a power-lawdistribution, so that N(N∗) dN∗
is the number of clusters with quantities of stars in
the range N∗to N∗+ dN∗:
This is due primarily to
N(N∗) dN∗= N−β
∗
dN∗
. (4)
We consider “clusters” containing a minimum of one
star, with the probability of larger clusters decreas-
ing with N∗according to the exponent β. Our models
use the RAN1 portable random number generator de-
scribed by Press etal. (1986), to generate the random
deviate x, having values between 0 and 1. The dis-
tribution in N∗given by equation 4 is then obtained
by
?
where N∗is assigned to the nearest integer. Since the
saturated end of the HII LF does not change slope,
as outlined in the previous section, we adopt β = 2 as
a typical value found for HII LFs in nearby galaxies
(e.g., KEH). Unlike MW97, we do not consider a dis-
tribution that is truncated at an upper limit in N∗for
our default models, although this condition is relaxed
in § 3.3.
For each cluster, we then draw N∗ stars from a
stellar mass distribution truncated above a mass limit
mup:
N∗=
−x
?
1 − β
??1/(1−β)
, (5)
n(m) dm ∝
??m
mup
?−γ
− 1
?
dm, (6)
where n(m) dm is the number of stars in the range
m to m + dm. These are generated with an algo-
rithm analogous to that for N∗. Equation 6 describes
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a power-law for m ≪ mup, and is the same ana-
lytic form used by MW97 for their N∗ distribution
and IMF; we refer to this form as a truncated power-
law, following their nomenclature. We use a Salpeter
(1955) power-law slope of γ = 2.35, and adopt an
upper-mass limit mup = 100M⊙. We include stars
down to a lower-mass limit mlo= 17M⊙(see below).
The Hα luminosity l associated with individual stars
is dependent on the mass and main-sequence lifetime
tms, for which we also use power law parameteriza-
tions:
m ∝ lδ
, (7)
normalized so that lup≡ l(mup) = 3 × 1038ergs−1;
and
m ∝ t−d
ms
, (8)
normalized so that tms(mup) = 2.8 Myr (Schaerer
etal. 1993). For stars in the range 17 < m < 100M⊙,
we find a value of d = 0.7 from Schaerer etal. (1993).
We caution the reader that the power-law represen-
tations of equations 7 and 8 are approximations; the
relations steepen significantly toward lower masses.
As discussed below, larger values of δ imply a greater
sensitivity of l to m, thereby increasing the range of L
in the unsaturated regime, for a given range of m. We
adopt δ ∼ 1.5, which yields a reasonable log llo∼ 37.3
for mlo= 17M⊙. For illustrative purposes, it is help-
ful to show a larger dynamic range of unsaturated
objects, though we caution that in reality the range
of these objects is likely to be somewhat smaller for
our mass limits, since the m − l relation flattens to-
ward higher masses (e.g., Leitherer 1990). We have
adopted the value of mloto coincide roughly with the
value at which the l − m relation turns over more
steeply, since stars below ∼ 17M⊙do not contribute
strongly to the ionizing fluxes. There is also a rela-
tive increase in tms(m) at lower masses, but not as
significant as the change in l(m). It is useful to bear
in mind that the representative values of δ and d will
increase toward lower masses.
We consider two different formulations for the stel-
lar ionizing rate q0 with time. The first simply ap-
proximates that q0 remains constant until tms and
is zero thereafter. The second formulation accounts
for evolution of q0 during the stellar main-sequence
phase. Figure 1 shows q0 for stars of different ini-
tial masses as a function of time, from the models of
Schaerer & de Koter (1997). These models suggest
that l remains fairly constant until a time after which
it decreases roughly along the relation,
l = lup
?
t
tms(mup)
?−ζ
, (9)
where ζ ∼ 5, and t is the age from birth. Given the
uncertainties in massive star evolution and ionizing
fluxes, we adopt the crude approximation of constant
l until the time of intersection with equation 9, af-
ter which we adopt l from that relation. One conse-
quence of this formulation is that stars with initially
different luminosities can have the same l during cer-
tain of their fading periods. Figure 1 suggests that
this may not be far from reality. We caution that
the assumption of coeval star formation within indi-
vidual nebulae is likely to be an oversimplification,
and that the stellar age spread is typically similar to
the main-sequence lifetime of the most massive stars
(e.g., Massey etal. 1995). Therefore, the early neb-
ular luminosity evolution could potentially increase
somewhat before the onset of fading. Specific model-
ing of star formation scenarios would be necessary to
investigate this effect.
Our adopted lower-mass cutoff introduces artifi-
cial fluctuations that can be seen in all our models
at low L. The quantized nature of the stellar ioniz-
ing sources, combined with the truncation of stellar
masses below mlo, causes artificial kinks in the dis-
tribution of nebular luminosities at low L. For ex-
ample, the most prominent of these features appears
at logL = 37.6 in our models, or 0.3 dex above the
lower-L cutoff of logllo= 37.3. This dip results from
the fact that all nebulae at L∼< 37.6 contain single
stars, selected according to the IMF, whereas two-star
nebulae begin to contribute above this luminosity. To
more clearly demonstrate the features in the HII LF,
we therefore include 104clusters in our simulations,
although this number is higher than is seen in most
nearby galaxies. Finally, we truncate our model HII
LFs below the minimum luminosity llo, which is that
due to a single star of mass mlo. Although objects
originating at higher L evolve to luminosities below
llo for cases incorporating main-sequence evolution,
the omission of objects originating below llorenders
the model populations incomplete below that value.
3.1.Single Burst
Figure 2 shows two models for the evolution of an
H II region population created in a single burst, with
δ = 1.5, d = 0.7, and 17 < m < 100M⊙. The left
5