# The Hyper-Stable Disc of UGC 8839

Jason E. Young1 and Miriam Eleazer1,2
1Astronomy Department, Mount Holyoke College, 50 College Street, 01075, United
States of America
2Astronomy Department, Wesleyan University, 45 Wyllys Avenue, 06459, United
States of America
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The low surface brightness (LSB) spiral UGC 8839 is nearly devoid of star
formation aside from a large $\rm H\text{\small II}$ region complex located in
the extreme outer disc. In order to understand the origin and nature of this
complex, we compare new $\rm H\alpha$ and archival broad-band images of UGC
8839 to similar data for four other spiral galaxies. We conclude that the
extreme off-axis star formation in UGC 8839 is likely due to a hyper-stable
disk that is dark matter dominated at all radii, with the Toomre parameter
reaching a minimum only in the extreme outer disc. Using analysis strategies
designed to be particularly insensitive to the pitfalls of low-surface
brightness objects and small number statistics, we determine that the presence
of this complex in UGC 8839 is not exceptional when the $\rm H\text{\small
II}$ region luminosity function is modelled by a power law, suggesting that it
is a native structure and not a merging satellite. However, we find that the
entire population of $\rm H\text{\small II}$ regions in UGC 8839 shows a
preference for larger galactocentric radii when compared to $\rm H\text{\small
II}$ regions in the other galaxies in our sample. UGC 8839 dramatically
highlights the relationship between the baryonic/dark matter ratio and disk
stability. A three-body interaction, similar to a scaled-down version of the
interaction suspected to be responsible for Malin 1, is consistent with the
extreme outer disk star formation that we see in the extended disk of UGC
8839.

###### keywords:

(ISM:) HII regions – galaxies: star formation – galaxies: evolution

††pubyear: 2015††pagerange: The Hyper-Stable Disc of UGC 8839–A

##  1 Introduction

Low surface brightness (LSB) spirals are extreme late-type spirals (Sd/Sm/Irr)
with blue optical colours and high gas fractions McGaugh et al. (1995); de
Blok et al. (1995); de Blok et al. (1996). Paradoxically, they also typically
exhibit low star-formation rates (e.g., van den Hoek et al., 2000; Kim, 2007).
This presents an interesting problem for our understanding of the ISM as the
bidirectional interface between galaxy dynamics and stellar populations. As a
particularly gas-rich but $\rm H\alpha$-faint object, the LSB spiral UGC 8839
is an excellent test case for ideas about the evolution of LSB spirals.

This paradox may be partly solved by sporadic star formation. Boissier et al.
(2008) find that LSB spirals have Galex FUV-NUV colours similar to but
slightly redder than HSB spirals, and discuss the possibility that the red
FUV-NUV colours are due to the fading of young but not actively forming
population. Using hydrodynamic simulations Vorobyov et al. (2009) are able to
replicate typical B-V colours and $\rm H\alpha$ equivalent widths of LSB
galaxies with sporadic star formation; irregular bursts keep the galaxy disc
comparatively blue, but the time-averaged star formation remains low. In our
earlier paper, Young et al. (2020), we reported on the spatially resolved
star-formation history of the LSB spiral UGC 628, and found evidence that
recent star formation seems to be sporadic, occasionally rising to a starburst
level.

However, we also found that the current/recent star formation in UGC 628 is
located almost entirely on the outer edge of the disc. This may be typical of
LSB spirals: In a sample of 1000 galaxies, Huang et al. (2013) showed that
edge-dominant starbursts are more common in LSB or otherwise late-type
galaxies, although few are as edge-dominated as UGC 628.

Edge-dominant star formation may also explain the comparatively flat
metallicity gradients reported in LSB galaxies. de Blok & van der Hulst (1998)
measure O/H ratios using nebular lines in a sample of three LSB galaxies, and
find very shallow and/or flat metallicity gradients. In Young et al. (2015) we
report a slightly inverted metallicity gradient in UGC 628. Bresolin &
Kennicutt (2015) find a plausible solution by pointing out that LSB spirals
also have shallower stellar light profiles; that is, longer exponential scale
lengths for their exponential discs. When metallicity gradients are calculated
in terms of scale length instead of physical units, HSB and LSB spirals have
similar metallicity gradients. If LSB spirals preferentially form stars near
edges of their discs, this would result in discs where both the stars and the
metals are more diffuse, and cleanly explain the longer scale lengths and
shallower metallicity profiles.

If indeed the characteristics of LSB spirals can be largely explained by
sporadic, edge-dominant star formation, then isolated LSB spirals that show
extreme off-centre star formation are cases that highlight the physical
differences between LSB and HSB spirals.

We can also gain some insight into the physical conditions which inhibit star
formation in gas-rich LSB spirals by looking at the $\rm H\text{\small II}$
region luminosity function (HRLF). Classical $\rm H\text{\small II}$ /
morphology relationships (e.g., Kennicutt et al., 1989) show that the HRLF
follows a steep power law in early-type spirals and a shallow power law in
late-type spirals. That is, early types favour many small $\rm H\text{\small
II}$ regions while late types favour a small number of larger $\rm
H\text{\small II}$ regions. As extreme late-types, LSB spirals might be
expected to have a shallow HRLF. However, unlike most high surface brightness
(HSB) late types, LSB spirals typically have low star-formation rates,
hampering attempts to measure the HRLF in individual LSB spirals.

Helmboldt et al. (2009) tackle this problem by stacking HRLFs from LSB and HSB
spirals, and then comparing the stacked HRLFs. They find that a key difference
is that the HRLF in HSB spirals is a Schechter function, which becomes steeper
above ${\rm log}L_{\rm H\alpha}{}\rm[erg\,s^{-1}]\\!=\\!38.6$. LSB spirals do
not seem to show this break. Overall low star-formation rates mean that LSB
spirals are less likely to host giant $\rm H\text{\small II}$ regions, but,
per unit star-formation rate, they are actually more likely to host giant $\rm
H\text{\small II}$ regions. If this is correct, then giant $\rm H\text{\small
II}$ regions with ${\rm log}L_{\rm H\alpha}{}{\rm[erg\,s^{-1}]}{\,\gtrsim
38.6}$ may be the sites of greatest contrast between HSB and LSB spirals, and
making cases such as UGC 8839 particularly worth studying.

Given these facts, the LSB spiral UGC 8839 is a very interesting case. With a
surface brightness of $\mu_{g}(0)=23.1\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$, it falls well within the LSB category,
though it is by no means an extreme example. We present $\rm H\alpha$ images
which show that UGC 8839 is largely devoid of star formation, with the
exception of a ${\rm log}L_{\rm H\alpha}{}{\rm[erg\,s^{-1}]}{=\\!39}$ $\rm
H\text{\small II}$ region complex located at the extreme outer edge of the
galactic disc. In this work, we examine the regions of this complex in the
context of the HRLF and the broad-band disc of UGC 8839. For comparison, we
examine three other LSB spirals and an HSB spiral in the same fashion.

In Section 2 we describe our targets list, our observations, and our reduction
methods. In Section 3 we develop a methodology to determine a probability
distribution for the power-law index of the HRLF in the limit of a small
sample size. We then use the constraints on the power-law index to show that
the giant $\rm H\text{\small II}$ region complex in UGC 8839 is likely drawn
from the same population as the remaining smaller $\rm H\text{\small II}$
regions in UGC 8839. In Section 4 we show that the entire population of $\rm
H\text{\small II}$ regions in UGC 8839 is located further out in the galactic
disc than the $\rm H\text{\small II}$ regions in our other sample galaxies. In
Section 5 we examine several interpretations and implications for other LSB
spirals.

##  2 Observations

###  2.1 Targets

Our targets are listed in Table LABEL:tbl:targets, and include four LSB
spirals and one HSB spiral. These targets were drawn from the catalogues of
LSB galaxies presented in McGaugh et al. (1995) and Kim (2007). Because
earlier studies of the HRLF have achieved reasonably complete samples down to
$\log{L_{\rm H\alpha}{}{\rm[erg\,s^{-1}]}}=37$ in galaxies out to 30Mpc (e.g.,
Caldwell et al., 1991), we restricted our targets to distances $\lesssim 30\rm
Mpc$.

UGC 8839: This galaxy is classified as Im, but close inspection reveals faint
extended spiral arms (Figure 1). The central surface brightness of UGC 8839 is
roughly a magnitude fainter than typical dark skies, making it decidedly an
LSB spiral. The large $\rm H\text{\small II}$ region complex comprised of
regions A, B, and C (Figure 2) appears to be an extension of the southern
spiral arm, and accounts for approximately 69% of the discrete star formation
in this galaxy. We will refer to the complex collectively as ABC, although the
individual regions will be analysed as discrete regions in the same manner as
the other regions in our sample. The identification of this complex, located
in the extreme outer disc, motivated this paper.

UGC 5633: This is a strongly barred galaxy, a rare characteristic for LSB
spirals, yet with a central surface brightness of $\mu_{0}(g)=\rm
22.9mag/{{}_{\Box^{{}^{\prime\prime}}}}$ UGC 5633 clearly falls within the LSB
category. In Figure 2 we see some $\rm H\alpha$ emission associated with this
bar, suggesting that it is a gaseous as well as a stellar bar.

UGC 6151: The $\rm H\alpha$ image for this object shows spirals arms traced
out by the $\rm H\text{\small II}$ regions. One of the $\rm H\text{\small II}$
regions is located at the exact centre of the disc. This central region is
likely the cause of the SDSS DR15 spectral classification of UGC 6151 as a
starburst galaxy; in fact the galaxy-wide $\rm H\alpha$ luminosity is not
particularly high (see Table LABEL:tbl:targets).

UGC 6181: This galaxy is a marginal LSB, with $\mu_{0}(g)=\rm
22.2mag/{{}_{\Box^{{}^{\prime\prime}}}}$. It hosts numerous star-forming
regions at the edge of the disc. The projected distance between UGC 6181 and
UGC 6151 is $\sim\\!640$kpc, close enough that they may be bound, although
there are no obvious signs of interaction.

NGC 4455: This HSB spiral is included in this study so that our sample would
span a range of surface brightnesses and chosen specifically because of its
proximity (only 10 Mpc) and availability during our observing run. It is a
late type spiral with an SBd morphology, although the classification is
dubious since it is inclined at $78^{\circ}$. The estimated mass is
log($\mathcal{M_{\star}}/\mathcal{M_{\odot}}$) = 11 (Kim, 2007).

Table 1: Target Galaxies and Observations | Typea | Da | $\rm g$ | $\rm M_{g}$ | $\mu_{0}(g)$ | $\rm r$ | $\rm M_{r}$ | $\mu_{0}(r)$ | $\rm z$ | $\rm M_{z}$ | $\mu_{0}(z)$ |  $\rm H\alpha$+0 |  $\rm H\alpha$+16 | Date
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Mpc | mag. | mag. | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | mag. | mag. | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | mag. | mag. | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | sec. | sec. | 03/2018
UGC 8839 | Im | 22.2 | 14.6 | -17.1 | 23.0 | 14.2 | -17.5 | 22.6 | 14.0 | -17.7 | 22.3 | 2120 | 1820 | 16,17
NGC 4455 | SBd | 10.4 | 12.8 | -17.3 | 20.8 | 12.5 | -17.6 | 20.4 | 12.3 | -17.8 | 20.2 | 480 | 160 | 16
UGC 6181 | Im | 23.7 | 14.2 | -17.7 | 22.2 | 13.8 | -18.1 | 21.8 | 13.1 | -18.8 | 21.5 | 3900 | 2400 | 16
UGC 6151 | Sm | 24.3 | 14.3 | -17.6 | 22.8 | 14.0 | -17.9 | 22.4 | 13.8 | -18.1 | 22.1 | 3900 | 2400 | 16
UGC 5633 | SBdm | 23.4 | 14.1 | -17.8 | 22.9 | 13.7 | -18.2 | 22.4 | 13.5 | -18.3 | 22.1 | 1620 | 1080 | 17
a NED |  |  |  |  |  |  |  |  |  |

###  2.2 Observations and Reduction

The $\rm H\alpha$ images in our study were collected with the Half Degree
Imager (HDI) on the WIYN 0.9 meter telescope at Kitt Peak National Observatory
on March 16-18 of 2018.

On-band observations were made using the $\rm H\alpha$+0nm filter
($\lambda=6547-6606$Å), and off-band observations were made using the $\rm
H\alpha$+16nm filter ($\lambda=6712-6772$Å). The data was reduced in a
standard fashion using biases and a combination of dome and twilight flats.
Because of the large field of view of the HDI, it was possible to identify
SDSS quasars with publicly available spectra in all of our target fields,
allowing for contemporaneous flux calibration. To do this, we convolved the
SDSS quasar spectra with our filter transmission curves (Ahumada et al.,
2020), and then adopted the average flux/count-rate ratios as our calibration
and the scatter in the ratios as the uncertainties on our calibration.

As a final calibration step, the flux calibrated images were then corrected
for foreground galactic extinction using the Cardelli extinction law (Cardelli
et al., 1989) and the E(B-V) values listed on NASA/IPEC Extragalactic Database
(NED), derived from the dust maps in Schlegel et al. (1998).

After calibration, the off-band $\rm H\alpha$+16nm images were subtracted from
the on-band $\rm H\alpha$+0nm images to remove the stellar continuum. Then, 2Å
equivalent width of stellar continuum was added back into the on-band images
to account for the stellar absorption underneath the $\rm H\alpha$ emission
line. The choice of 2Å equivalent width as an estimation of the strength of
the Balmer absorption lines (in the absence of high resolution spectra) is
motivated by the careful analysis of $\rm H\alpha$ images of $\rm
H\text{\small II}$ regions in McCall et al. (1985) and Oey & Kennicutt (1993),
of $\rm H\beta$ images of LSB galaxies in Kuzio de Naray et al. (2004), and of
$\rm H\beta$ and $\rm H\gamma$ data in our earlier work, Young et al. (2015).
The average fractional correction to the $\rm H\alpha$ fluxes due to 2Å of
continuum absorption was $0.054^{+0.051}_{-0.027}$. The average fractional
correction for regions in UGC 8839 was even smaller,
$0.007^{+0.004}_{-0.003}$. Note that these corrections are generally much
smaller than the measurement uncertainties listed in Table 3, and have
relatively little impact on the analysis below. As discussed in Section 2.3,
uncertainties in the continuum fluxes were propagated through the continuum
removal and the 2Å correction in quadrature with the uncertainties in the on-
band $\rm H\alpha$ fluxes. No attempt was made to remove the [N
ii@]$\lambda$6584 line from the $\rm H\alpha$ images.

Figure 1:  $\rm H\alpha$ image of UGC 8839, overlaid with SDSS g-band surface
brightness contours. The contours are labelled in units of $\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$.

In Figure 1 we show the $\rm H\alpha$ image of UGC 8839, overlaid with SDSS
DR16 g-band surface brightness contours (Ahumada et al., 2020). The $\rm
H\alpha$ image is nearly blank, aside from the bright spot located in the
lower right, the ABC complex. The identification of individual $\rm
H\text{\small II}$ regions will be discussed in Section 2.3. Curiously, UGC
8839 seems to be nearly devoid of star formation, aside from this large clump
at the extreme edge of the disc. Although UGC 8839 is classified as an Im
type, the g-band contours do show faint spiral arms. Interestingly, the ABC
complex appears to be an extension of the south western arm.

###  2.3 Identification and Measurement of $\rm H\text{\small II}$ Regions

The calibrated images were visually inspected for $\rm H\text{\small II}$
regions using SAOImage DS9, and cross-checked against publicly available sky
surveys (SDSS DR16, 2MASS, DESI Legacy Imaging Survey). $\rm H\text{\small
II}$ regions were enclosed with a circular or elliptical boundary in SAOImage
DS9 using the shape tool. Region clusters were separated into individual
regions where definition between regions could be determined in both archival
survey images and in our $\rm H\alpha$ images. These region boundaries were
identified by careful inspection of the images to ensure that pixel brightness
reached a minimum between regions. The $\rm H\alpha$ images, along with the
identified $\rm H\text{\small II}$ regions, are shown in Figure 2. The fluxes
within the regions are listed in Table 3. The uncertainties in the fluxes were
derived by propagating in quadrature the calibration uncertainties and the
standard deviation of the background (empty) pixels in the $\rm H\alpha$+0nm
and $\rm H\alpha$+16nm images through the processes of continuum removal and
the correction for the 2Å equivalent width of stellar absorption underneath
the $\rm H\alpha$ emission lines, as discussed in Section 2.2.

Additionally, whole-galaxy photometry was performed on the $\rm H\alpha$
images. All of the galaxies in our sample had at least some diffuse $\rm
H\alpha$ emission not clearly associated with any individual region. The whole
galaxy measurements and diffuse fractions are listed in Table 3.

It is worth noting that Richards et al. (2016) report a whole-galaxy $\rm
H\alpha$ flux for UGC 8839 as $\log{f_{\rm H\alpha}{}[\rm
erg\,s^{-1}cm^{-2}]}=-13.61\pm 0.18$, which differs from our value by more
than $2\times$. This difference is driven by the fact that the $\rm H\alpha$
photometry in Richards et al. (2016) is confined to $R_{25}$ as measured in
3.6µm Spitzer IRAC images. In the case of UGC 8839,
$R_{25}=34^{\prime\prime}\\!\\!.1$. All of the $\rm H\text{\small II}$ regions
we report here are beyond this radius, and, as will be shown in subsequent
sections, the $\rm H\alpha$ light in UGC 8839 is strongly biased toward the
extreme outer disk.

Figure 2:  Sample galaxies, DESI Legacy Imaging Survey grz images (top) and
$\rm H\alpha$ images with $\rm H\text{\small II}$ regions marked (bottom). The
angular scale is the same in both sets of images. The $\rm H\alpha$ image of
UGC 6151 has bright feature in the upper right corner; this is asteroid, and
can be ignored.

###  2.4 Broad-Band Surface Photometry

In order to examine the significance of the location of the ABC complex on the
outer edge of the disc of UGC 8839, Section 4 discusses the local broad-band
surface brightness around each $\rm H\text{\small II}$ region within each of
our sample galaxies. These surface brightness measurements were made using the
publicly available DESI Legacy Imaging Survey g,r,z-band images and are listed
in Table 3.

Figure 3: A close-up of the grz Legacy Sky Survey Image of UGC 8839 showing
several $\rm H\text{\small II}$ regions. The dashed shapes were used for
photometry in the $\rm H\alpha$ images, and the solid annuli were used for
surface photometry in the Legacy broad-band images.

Because the forthcoming analysis in Section 4 relies on the local g,r,z-band
surface brightness around each region being representative of the “typical”
disc brightness at that galactocentric radius, it is necessary to exclude the
$\rm H\text{\small II}$ regions themselves from the local surface brightness
calculation. After region identification (Section 2.3), a circular annulus
with an inner radius of $1^{\prime\prime}\\!\\!.3$ outside the $\rm
H\text{\small II}$ region and a width of $1^{\prime\prime}\\!\\!.3$ was placed
on each region. Adjustments by hand were made as necessary, and nearby clumps
or local structures were masked out to ensure that the annulus sampled the
true local disk. We also tuned the annular regions to be large enough that
arm/inter-arm variations in surface brightness were averaged out. An example
is shown in Figure 3. All unmasked pixels in the broad-band images that were
within each region’s annulus and not within the photometry region of any other
$\rm H\text{\small II}$ region were averaged to generate a local surface
brightness around each $\rm H\text{\small II}$ region. This method is similar
to the method used to determine the local surface brightness near $\rm
H\text{\small II}$ regions in Helmboldt et al. (2009).

The central g,r,z-band surface brightness of each galaxy was measured as the
average flux within a circular region placed at the visible centre (see green
circular regions in Figure 2). When possible, this circle was given a 5″
radius. In NGC 4455 this was not possible because of the high density of
bright knots (likely due to the high inclination of this galaxy). The bright
$\rm H\text{\small II}$ region near the centre of UGC 6151 and the bar in UGC
5633 also posed problems. Both of these bright objects were excluded (shown as
red regions in Figure 2). In UGC 5633, the green circle was expanded to 15″.

All surface brightnesses were corrected for foreground galactic extinction
using $A_{\lambda}$ values listed on NED. We adopted the standard deviation of
background (empty) pixels in the g,r,z-band images as uncertainties in the
fluxes of individual galaxy pixels, and propagated those uncertainties forward
through the surface brightness calculations. These uncertainties are listed in
Table 3. Note that these uncertainties only account for random statistical
uncertainties, and not systematic calibration uncertainties. Because the
analysis in Section 4 relies entirely on $\mu-\mu_{0}$, the effects of
calibration uncertainties do not apply.

##  3 Luminosity Distribution

The left panels in Figure 4 show luminosity histograms for the $\rm
H\text{\small II}$ Regions in our sample galaxies. The histograms taper off
slowly at high luminosities and abruptly at low luminosities. The upper end of
the curve is likely real and representative of the HRLF, but the lower end of
the curve is driven by incompleteness in our observations. The completeness
limit, $L_{c}$, marked by the turnover point, is set by S/N in the images and
also by the distances from Earth. Table LABEL:tbl:statistics lists the
measured $\rm H\alpha$ luminosity completeness limit ($L_{c}$) for each galaxy
in our sample.

In order to test the hypothesis that the large $\rm H\text{\small II}$ regions
comprising the ABC complex in UGC 8839 are actually unusual, we will quantify
the likelihood that UGC 8839 would host such regions by using the $\rm
H\text{\small II}$ region luminosity function (HRLF) as a likelihood
predictor.

The primary challenge in fitting even a two parameter distribution to LSB
galaxies is the scarcity of regions, often too few for standard “best fit"
methods given random variation. One solution is to stack the distributions
from a sample of LSB galaxies and in Helmboldt et al. (2009). This kind of
technique is effective only in determining group-wide characteristics, and is
not an ideal choice in this case since we are aiming to test the hypothesis
that UGC 8839 has aberrant characteristics.

Instead, we organise the $\rm H\text{\small II}$ regions of each galaxy into
histogram bins, and then derive the probability $P(\geq\\!N_{big}|H)$ that
galaxy each will be able to populate its highest bin with $N_{big}$ regions
given the distribution of the lower bins. We start in Section 3.1 by
calculating $\frac{dP(a|H)}{da}$, the probability density of the power-law
index $a$ given that galaxy’s observed histogram $H$ (excluding the highest
bins). Then, in Section 3.2 we use $\frac{dP(a|H)}{da}$ to calculate
$P(\geq\\!N_{big}|H)$. If a galaxy’s highest populated bin is an extreme
outlier compared to the others, then $P(\geq\\!N_{big}|H)$ will be very small,
indicating that the most luminous regions are not drawn from the same
population. A more moderate value of $P(\geq\\!N_{big}|H)$, near 50%, would
indicate that the most luminous regions can be fully explained by the same
HRLF that the smaller regions are drawn from.

Figure 4: Left: $\log{L_{\rm H\alpha}{}}{\rm[erg\,s^{-1}]}$ histograms and
smoothed density functions of the $\rm H\text{\small II}$ regions in each
galaxy. The vertical dashed line marks the peak of the smoothed density
function, which we adopt as the completeness limit $L_{C}$. Right: Probability
density functions for the power-law index $a$. Although the histograms shown
on the left are binned with $\Delta\log{L}{\rm[erg\,s^{-1}]}=0.3$, the
probability density functions have been calculated for
$\Delta\log{L}{\rm[erg\,s^{-1}]}=0.3$ (solid curve) and
$\Delta\log{L}{\rm[erg\,s^{-1}]}=0.2,0.4$ (dashed curves). Note that they are
not appreciably different, showing that the choice of bin width does not
significantly impact our results. The $1\sigma$ limits on $a$ (for the
$\Delta\log{L}{\rm[erg\,s^{-1}]}=0.3$ curves) are marked with dashed vertical
lines in the right panels, and the expectation curves for those limits are
superimposed on the histograms in the left panels as solid curves. The
histograms generally follow the expectation curves.

###  3.1 Determination of the Power Law Index

We start from the premise that the HRLF follows by a power law:

$dN=AL^{a}dL$ (1)

where $A$ is a normalisation constant and $a$ is the power-law index. This
power law must break down at low luminosities for physical reasons, however in
practice the lower limit of its applicability is set by the completeness
limit, $L_{C}$. For our analysis, we adopt the luminosity of the peak
histogram bin as $L_{C}$.

We divide the luminosity range above $L_{C}$ into $B$ log luminosity bins
(just as a histogram with $B$ bins). Figure 4 full shows histograms with a
binning of $\Delta\log{L_{\rm H\alpha}{}{\rm[erg\,s^{-1}]}}=0.3$, with $L_{C}$
marked as a vertical dashed line. We have repeated the calculations which
follow using $\Delta\log{L_{\rm
H\alpha}{}{\rm[erg\,s^{-1}]}}=0.2\text{\,and\,}0.4$. The results from all
three sets of calculations are shown in Table LABEL:tbl:statistics. Later in
this section we demonstrate that the exact binning does not impact our general
conclusions. If we consider Equation 1 in the context of a single $\rm
H\text{\small II}$ region as a probability that that region will have a exact
luminosity $L$, then the probability that that region will fall into bin $b$
is:

$P_{b}=\int^{L_{b+1}}_{L_{b}}AL^{a}dL=\frac{A}{a+1}\left(L_{b+1}^{a+1}-L_{b}^{a+1}\right)$
(2)

where $A$ is defined to normalise the total probability.

Let $N$ be the total number of regions, and $n_{b}$ be the number of regions
in bin $b$. Since each of the $n_{b}$ regions has the same expression for
$P_{b}$, we can express the joint probability that all $n_{b}$ regions will
fall within bin $b$ as:

$P_{b}(n_{b})=\binom{N}{n_{b}}\left(P_{b}\right)^{n_{b}}=\binom{N}{n_{b}}\left(\frac{A}{a+1}\right)^{n_{b}}\left(L_{b+1}^{a+1}-L_{b}^{a+1}\right)^{n_{b}}$
(3)

Note the presence of $\binom{N}{n_{b}}$ to account for number of combinations,
which count identically.

Finally, we multiply all the $P_{b}(n_{b})$ together to find the joint
probability that all the regions will fall within their respective bins to
form the histogram $H$ given a value of the power-law index $a$:

$P(H|a)=\prod\limits_{b=0}^{B}P_{b}(n_{b})=\prod\limits_{b=0}^{B}\binom{N_{b}}{n_{b}}\left(\frac{A}{a+1}\right)^{n_{b}}\left(L_{b+1}^{a+1}-L_{b}^{a+1}\right)^{n_{b}}$
(4)

Here $N_{b}=N-\sum\limits_{b^{\prime}=0}^{b-1}$$n_{b}$, the number of regions
remaining after removing those falling into bins below bin $b$. Since the
total number of regions $N=\sum\limits_{b=0}^{B}n_{b}$ we can write:

$P(H|a)=\left(\frac{A}{a+1}\right)^{N}\prod\limits_{b=0}^{B}\binom{N_{b}}{n_{b}}\left(L_{b+1}^{a+1}-L_{b}^{a+1}\right)^{n_{b}}$
(5)

The expression above assumes a value for $a$, when in fact $a$ is uncertain.
The differential probability of observing histogram $H$ is the joint
probability of observing a histogram $H$ given $a$ and the differential
probability observing the value of $a$:

$dP(H|a)=\left(\frac{A}{a+1}\right)^{N}\prod\limits_{b=0}^{B}\binom{N_{b}}{n_{b}}\left(L_{b+1}^{a+1}-L_{b}^{a+1}\right)^{n_{b}}dP(a)$
(6)

Then, by Bayes’ Theorem:

$dP(a|H)=\frac{P(H|a)}{P(H)}dP(a)=\frac{P(H|a)}{P(H)}\frac{dP(a)}{da}da$ (7)

To find $P(H)$, the total probability of observing the histogram $H$
regardless of the value of $a$, we integrate over all possible values of $a$:

$P(H)=\int_{a_{min}}^{a_{max}}\frac{dP(H|a)}{da}da$ (8)

In our calculations we use the integral bounds $-4\\!<\\!a\\!<\\!-1$, however
the exact choice of values for the integral bounds does not have an impact on
the integral so long as $P(H|a)$ goes to zero for $a<-4$ and $a>-1$. The final
plots of $\frac{dP(a|H)}{da}$, shown in the right panels of Figure 4, go to
approximately zero for these limits, demonstrating that $P(H|a)$ must also go
zero for these limits.

Likewise, the choice of prior is $\frac{dP(a)}{da}$ becomes moot for extreme
values of $a$ since it is being multiplied by $P(H|a)$. For moderate values of
$a$ the prior does have an impact. We assume a flat prior
$\frac{dP(a)}{da}=1$; although a shallow power-law might be expected for late-
type spirals, the HRLF is not well studied in LSB galaxies, and any strong
assumptions on the prior would have a poor foundation.

Applying $P(H)$ and our flat prior yields a final expression for the
probability density:

$dP(H|a)=\frac{\left(\frac{A}{a+1}\right)^{N}\prod\limits_{b=0}^{B}\binom{N_{b}}{n_{b}}\left(L_{b+1}^{a+1}-L_{b}^{a+1}\right)^{n_{b}}da}{\int_{a_{min}}^{a_{max}}\left(\frac{A}{a+1}\right)^{N}\prod\limits_{b=0}^{B}\binom{N_{b}}{n_{b}}\left(L_{b+1}^{a^{\prime}+1}-L_{b}^{a^{\prime}+1}\right)^{n_{b}}da^{\prime}}$
(9)

We have applied this expression to the histograms shown in the left panels in
Figure 4 to determine the probability density for the HRLF power law index
$a$, shown in the right panels in Figure 4.

The solid curves show the probability density derived using bin widths of
$\Delta\log{L_{\rm H\alpha}{}}{\rm[erg\,s^{-1}]}=0.3$, the dashed curves using
$\Delta\log{L_{\rm H\alpha}{}}{\rm[erg\,s^{-1}]}=0.2$ and $0.4$. The black
curves were generated excluding the highest populated bins, and the brown
curves were generated using all the histogram bins. Note that the exclusion of
the highest populated bin causes the curves to be broader (more uncertainty)
and also skews the curves to steeper (lower) powers. The increase in
uncertainty results from the diminished data set. The skew results from the
fact that it is exclusively the high end data points being removed. In the
analysis that follows we use the distribution with the highest luminosity bins
removed (solid black curves) because our goal is to calculate the probability
that each galaxy would host its largest regions given the distribution of the
smaller regions. That said, all the curves are generally similar to each
other, indicating that these assumptions do not have a significant impact on
our analysis.

Note also that these curves, which are essentially normalised $P(H|a)$, go to
zero within the $-4\\!<\\!a\\!<\\!-1$ bounds, justifying our earlier
assumptions about the range of $a$.

The dashed vertical lines in the right panels of Figure 4 show the upper and
lower $1\sigma$ bounds for the $\Delta\log{L_{\rm
H\alpha}{}}{\rm[erg\,s^{-1}]}=0.3$ curves. The $1\sigma$ bounds and maximum
probability values for all three probability curves are listed in Table
LABEL:tbl:statistics. Although the histograms in the left panels of Figure 4
are sparsely populated, all of the power-law indices are constrained to within
$\pm 0.4$ dex.

Using the $1\sigma$ limits as the upper and lower bounds on the power-law
index, we have calculated upper and lower expectation curves for the number of
$\rm H\text{\small II}$ regions in each of the histogram bins, and over
plotted those on top of the histograms as solid curves. Matching the colour
scheme from the right panels, the black curves exclude the highest populated
bins and the brown curves include them. Both sets of expectation curves match
this histograms reasonably well, with random variation from small number
counting noise. We conclude that the HRLF is well modelled by a power law
based on the match between the expectation curves and the histograms.

Table 2: HRLF Properties |  |  |  |  | largest included | largest excluded
---|---|---|---|---|---|---
Galaxy | Bin Width | $L_{C}$ | $N\\!>\\!L_{C}$ | $N_{big}$ | $a$ | $P\left(N_{big}\right)$ | $a$ | $P\left(N_{big}\right)$
| dex | erg s-1 |  |  |  |  |  |
UGC8839 | 0.2 | 37.50 | 15 | 2 | $-1.85^{+0.2}_{-0.3}$ | 0.58 | $-2.04^{+0.3}_{-0.4}$ | 0.39
UGC8839 | 0.3 | 37.50 | 15 | 2 | $-1.91^{+0.3}_{-0.3}$ | 0.62 | $-2.13^{+0.4}_{-0.4}$ | 0.41
UGC8839 | 0.4 | 37.50 | 15 | 2 | $-1.93^{+0.3}_{-0.3}$ | 0.70 | $-2.14^{+0.4}_{-0.4}$ | 0.52
NGC4455 | 0.2 | 37.80 | 20 | 1 | $-2.02^{+0.3}_{-0.3}$ | 0.80 | $-2.12^{+0.3}_{-0.3}$ | 0.73
NGC4455 | 0.3 | 37.80 | 20 | 3 | $-2.00^{+0.3}_{-0.3}$ | 0.45 | $-2.35^{+0.4}_{-0.5}$ | 0.18
NGC4455 | 0.4 | 37.80 | 20 | 3 | $-2.06^{+0.3}_{-0.3}$ | 0.51 | $-2.46^{+0.4}_{-0.5}$ | 0.22
UGC6181 | 0.2 | 37.50 | 22 | 2 | $-1.89^{+0.2}_{-0.3}$ | 0.72 | $-2.03^{+0.3}_{-0.3}$ | 0.57
UGC6181 | 0.3 | 37.50 | 22 | 3 | $-1.88^{+0.2}_{-0.3}$ | 0.64 | $-2.08^{+0.3}_{-0.3}$ | 0.40
UGC6181 | 0.4 | 37.50 | 22 | 4 | $-1.97^{+0.2}_{-0.3}$ | 0.47 | $-2.38^{+0.4}_{-0.5}$ | 0.16
UGC6151 | 0.2 | 37.50 | 23 | 1 | $-2.05^{+0.3}_{-0.3}$ | 0.68 | $-2.16^{+0.3}_{-0.3}$ | 0.57
UGC6151 | 0.3 | 37.50 | 23 | 1 | $-2.07^{+0.3}_{-0.3}$ | 0.66 | $-2.20^{+0.3}_{-0.4}$ | 0.54
UGC6151 | 0.4 | 37.50 | 23 | 1 | $-2.10^{+0.3}_{-0.3}$ | 0.62 | $-2.26^{+0.3}_{-0.4}$ | 0.49
UGC5633 | 0.2 | 37.50 | 33 | 1 | $-1.92^{+0.2}_{-0.2}$ | 0.89 | $-1.98^{+0.2}_{-0.2}$ | 0.85
UGC5633 | 0.3 | 37.50 | 33 | 1 | $-1.88^{+0.2}_{-0.2}$ | 0.91 | $-1.93^{+0.2}_{-0.2}$ | 0.88
UGC5633 | 0.4 | 37.50 | 33 | 1 | $-1.91^{+0.2}_{-0.2}$ | 0.90 | $-1.97^{+0.2}_{-0.2}$ | 0.85

###  3.2 Likelihood of 1st Ranked Bin

In order to determine whether the ABC complex is extraordinary given the
entire population of $\rm H\text{\small II}$ regions in UGC 8839, we will
derive a general formula for the probability $P(\geq\\!N_{big})$ that a galaxy
with $N$ $\rm H\text{\small II}$ regions and a probability density function
$\frac{dP(a|H)}{da}$ will have at least $N_{big}$ large regions, where “large”
is defined as the most luminous populated histogram bin. This probability is a
measure of the HRLF’s ability to explain the presence of the most luminous
$\rm H\text{\small II}$ regions. If a galaxy’s most luminous populated bin is
an extreme outlier then $P(\geq\\!N_{big})$ will be very small, indicating
that it is very _unlikely_ that the most luminous regions in that galaxy are
drawn from the same population as the less luminous ones.

As long as $a\\!<\\!-1$, we can take the upper limit in Equation 2 to infinity
to find the total probability of a given $\rm H\text{\small II}$ region having
a luminosity greater than $L$:

$P(\geq\\!L)=-\frac{AL^{a+1}}{a+1}$ (10)

Note that this expression is positive despite the negative sign since $a+1<0$.

Since we are considering only $\rm H\text{\small II}$ regions with $L_{\rm
H\alpha}{}>{L_{c}}{}$, we can normalise to $P(\geq\\!L_{L_{c}}{})=1$, and let
$L=L_{big}$, the threshold luminosity for an $\rm H\text{\small II}$ region to
be included in the largest histogram bin:

$P(\geq\\!L_{big}|a)=\left(\frac{L_{big}}{{L_{c}}{}}\right)^{a+1}$ (11)

Given this expression for the probability of an individual $\rm H\text{\small
II}$ region having $L_{\rm H\alpha}{}>L_{big}$, we now consider a galaxy with
$N$ $\rm H\text{\small II}$ regions. The joint probability of exactly
$N_{big}$ regions all having luminosities $L_{\rm H\alpha}{}>L_{big}$ can be
expressed as the joint probability that $N_{big}$ regions will have $L_{\rm
H\alpha}{}>L_{big}$ and $N-N_{big}$ regions will have $L_{\rm
H\alpha}{}<L_{big}$, multiplied by $\binom{N}{N_{big}}$, the number of
combinations of $N_{big}$ regions drawn from a sample of $N$:

$P(N_{big}|a)=\binom{N}{N_{big}}\left(\frac{L_{big}}{{L_{c}}{}}\right)^{(a+1)N_{big}}\left(1-\left(\frac{L_{big}}{{L_{c}}{}}\right)^{a+1}\right)^{N-N_{big}}$
(12)

Note that if we sum $P(N_{big})$ over $N_{big}\\!\\!=\\!\\!0$ to
$N_{big}\\!\\!=\\!\\!N$, the sum is unity.

However, this expression for $P(N_{big}|a)$ is contingent upon the value of
the power-law index $a$, which is itself uncertain. We can express this by
using Equation 12 to find the joint probability of a power-law index with a
value of exactly $a$ and, given that value, also having exactly $N_{big}$
regions with luminosities $L_{\rm H\alpha}{}>L_{big}$:

$P(N_{big})=P(N_{big}|a)P(a)$ (13)

As discussed in Section 3.1, the number of $\rm H\text{\small II}$ regions in
our sample galaxies is too few for standard “best fit" methods given random
variation, so instead we derived a probability density function
$\frac{dP(a|H)}{da}$ given the observed histogram $H$. Using that result, Our
expression for $P(N_{big})$ then becomes a differential probability

$dP(N_{big})=P(N_{big}|a)dP(a|H)=P(N_{big}|a)\frac{dP(a|H)}{da}da$ (14)

Which then becomes an integral:

$P\left(N_{big}\right)=\int_{a_{min}}^{a_{max}}P(N_{big},a)\,\frac{dP(a|H)}{da}\,da$
(15)

This is is the total probability that, given the histogram $H$, a galaxy will
have exactly $N_{big}$ regions in its 1st ranked bin. However, in our case, we
are interested in testing the hypothesis that the 1st ranked bin is
anomalously overpopulated. To do this, we will calculate the probability that
the 1st ranked bin will be populated with at least the number of regions as in
our measurements:

$P\left(\geq\\!N_{big}\right)=1-\sum\limits_{N_{big}^{\prime}=0}^{N_{big}}P(N_{big}^{\prime})$

Using this formula, we present in Table LABEL:tbl:statistics the calculated
values of $P\left(\geq\\!N_{big}\right)$, the probabilities that the 1st
ranked luminosity bins would be populated with at least $N_{big}$ regions
given the total number of regions $N$ with $L_{\rm H\alpha}{}\\!>\\!L_{C}$ and
the probability density $\frac{dP(a|H)}{da}$. These results will be discussed
further in Section 5, however we find that the probability that UGC 8839 will
be able to populate its 1st ranked bin with the observed $N_{big}$ is 40-50%
for all choices of bin width. We conclude that presence of the ABC complex is
not extraordinary given entire observed population of $\rm H\text{\small II}$
regions in UGC 8839, and that a single power-law HRLF is fully capable of
explaining presence of the ABC complex in UGC 8839. The implications of this
conclusion will be discussed in greater depth in Section 5, however the main
point is that, although the ABC complex visually stands out and seems “out of
place” in UGC 8839 (Figure 1), it is actually in-line with the entire
population of $\rm H\text{\small II}$ regions.

##  4 Spatial Distribution

In Section 3 we showed that the presence of the ABC complex in UGC 8839 is not
extraordinary. The other striking aspect of the ABC complex is its location:
this complex, which accounts for 69% of the discrete $\rm H\alpha$ light, is
located 8 kpc from the galaxy’s centre, beyond the main body of the visible
disc (Figure 2). To explore the seemingly odd placement for such a large
region in an otherwise quiet galaxy, we quantify the spatial distribution of
star formation in the remaining galaxies in our sample and compare them to UGC
8839.

When comparing the spatial distribution of $\rm H\text{\small II}$ regions in
different galaxies, it is essential that we compare them on equal footing. For
example, the ABC complex’s location of 8 kpc from the centre of UGC 8839 puts
it in the extreme outer disk of this small galaxy. In comparison, 8 kpc is
roughly the solar circle in the larger Milky Way.

Instead of comparing the spatial distribution in terms in physical distances,
it is more sensible to compare them using a characteristic distance. Since
these galaxies are all late-types, the exponential scale length is a natural
length to use, defined by:

$\mu=\mu_{0}+\frac{2.5}{\ln{10}}\frac{r}{\alpha}$ (16)

Here $r$ is the deprojected angular distance from the centre of the galaxy,
$\alpha$ is the angular exponential scale length of the galaxy, and $\mu$ and
$\mu_{0}$ are the local and central surface brightnesses in $\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$. Expressing the position of the $\rm
H\text{\small II}$ regions within their respective galaxies in terms of
$\frac{r}{\alpha}$ rescales the galactocentric distances to the size of each
galaxy. Also, since both measurements are in angular units, any uncertainties
or systematic effects in the heliocentric distances are excluded. Note that
many galaxies are known to have piece-wise light profiles or “broken
exponential” profiles, and we caution against over interpretation of
$\frac{r}{\alpha}$ as a literal number of scale lengths. Instead, we are using
it here as a characteristic distance.

However, the practical application of Equation 16 requires the fitting of the
disc profile, which would inject a significant amount of uncertainty given the
faintness of our galaxies. For example, the deprojected angular positions of
the $\rm H\text{\small II}$ regions ($r$) would be heavily influenced by
uncertainties in the inclination angle and the position angle of their host
galaixes. Even discounting these issues, a straightforward application of
Equation 16 would be cumbersome for galaxies with broken exponential profiles.

Instead, we use $\left(\mu-\mu_{0}\right)\frac{\ln{10}}{2.5}$, which is
equivalent to $\frac{r}{\alpha}$ but sidesteps uncertainties about disc
geometry. As discussed in Section 2.4, $\mu$ was measured locally around each
$\rm H\text{\small II}$ region, and $\mu_{0}$ was measured in the geometric
centre of each galaxy. Because both measurements are local, no corrections for
inclination or geometry are needed. Also, because an annulus was used to
measure the local surface brightness around each region, the $\mu$ values for
each region do not reflect the characteristics of that region, but instead
reflect the characteristics of the disc at each galactocentric radius.

Azimuthal variations in $\mu$ from spiral arms as well as general clumpiness
pose a concern with our use of $\mu-\mu_{0}$ as a proxy for
$\frac{r}{\alpha}$. As discussed in Section 2.4, the annuli were chosen in
such a way to mitigate these issues. Additionally, our analysis uses three
filters, g, r, and z, which provides a secondary check: any effect seen only
or primarily in the g-band would be suspect on the grounds that galaxies are
clumpier and have higher arm/inter-arm contrast in bluer bands. Conversely,
any conclusions drawn based on all three bands, especially the redder bands,
are unlikely to be due to arms or clumpiness associated with star formation.

Figure 5:  All the $\rm H\text{\small II}$ regions in our study, plotted as
fractions of total $\rm H\alpha$ luminosity versus relative distance from the
galaxy centre. The plots on the left calculate the total luminosity as the sum
of the discrete $\rm H\text{\small II}$ regions, while the plots on the right
include the diffuse emission. In both cases and in all filters the regions
from UGC 8839 fall on the upper right envelope of the plot. Star formation in
UGC 8839 shows a clear preference for the disc edge.

Figure 5 shows all the $\rm H\text{\small II}$ regions in our sample plotted
by their fractional contribution to the total $\rm H\alpha$ light versus
$\left(\mu-\mu_{0}\right)\frac{\ln{10}}{2.5}$ for the g, r, and z filters. In
the left panels, the total $\rm H\alpha$ light is the sum of all the discrete
$\rm H\text{\small II}$ regions. In the panels on the right, the total $\rm
H\alpha$ light includes the diffuse emission. Note that the overall pattern of
points in the left and right panels is not appreciably different.

In all panels most of the $\rm H\text{\small II}$ regions fall within a
general cloud which is not significantly different from galaxy to galaxy, with
the notable exception of UGC 8839. The regions associated with UGC 8839 lie
primarily on the upper right envelope of the distribution. For their
luminosities, the $\rm H\text{\small II}$ regions in UGC 8839 are almost all
located farther from the centre than analogous region in the remaining
galaxies. The remaining galaxies show no discernible patterns. We find that
star formation in UGC 8839 shows a marked proclivity for the extreme outer
disk. The implications of disc-edge dominant star formation in UGC 8839 will
be discussed further in Section 5.4.

Note that the main effect, that the $\rm H\text{\small II}$ regions in UGC
8839 are preferentially on the disk edge (at larger $\frac{r}{\alpha}$), is
strongest in the r-band and still very strong in the z-band, where the
galaxies are the smoothest and the least clumpy. This shows that that
asymmetric features such as a arms and clumps do not contribute significantly
to our conclusions.

The consistency of this effect across filters also addresses another potential
pitfall: that errors in the measurement of $\mu_{0}$ would shift all the
points for a given galaxy left or right. Since the measurements in the
different filters are independent, we can eliminate this possibility.
Additionally, the difference between the points for UGC 8839 and the remaining
galaxies is approximately one half to one magnitude, far larger than our
measurement errors in $\mu_{0}$.

##  5 Discussion

###  5.1 General Comments on the HRLF

The power law indices in table Table LABEL:tbl:statistics are all around $-2$.
This consistent with expectations, since most late-type HSB galaxies are best
characterised by power laws in the range $-2\\!<\\!a\\!<\\!-1.5$ (e.g.,
Kennicutt et al., 1989).

The fraction of $\rm H\alpha$ light diffusely associated with each galaxy and
not with identifiable regions spans a range, from 27% in UGC 6151 to 81% in
UGC 6181. All except UGC 6151 and the HSB NGC 4455 have diffuse fractions
greater than 40%, generally in agreement the findings in O’Neil et al. (2007),
which show that LSB galaxies tend to have higher diffuse fractions of $\rm
H\alpha$ light than HSB galaxies, usually above 50%. It is not possible for us
to determine what fraction of the diffuse light may be unresolved $\rm
H\text{\small II}$ regions below our completeness limit $L_{C}$, however the
the consistent power law of $-2$ despite a $>\\!3\times$ variation in the
diffuse fraction suggests that this effect is not significantly impacting our
results. The southern end of the bar in UGC 5633 is even evident in the $\rm
H\alpha$ images as diffuse emission, although it only contributes about 5% to
the total diffuse emission.

###  5.2 The ABC Complex as an Intruder

One possible explanation for the presence of the ABC complex in the extreme
outer disc UGC 8839 is that the complex is an intruder object, perhaps a small
gas-rich dwarf merging with UGC 8839. For the following reasons, the analysis
above shows that this interpretation is unlikely:

First, in Section 3 we examined the probability that UGC 8839 would host its
largest regions given the population of smaller $\rm H\text{\small II}$
regions, and found that the probability is $\gtrsim\\!40\%$. While the
uncertainties due to low number statistics are significant, we can safely
conclude that the existence of the ABC complex in UGC 8839 is not, by itself,
extraordinary. The HRLF of UGC 8839 is fully capable of accounting for the ABC
complex, without requiring that the complex be an intruder object.

Second, in Section 4 we examined the location of $\rm H\text{\small II}$
regions in our sample galaxies by plotting the fractional contribution of each
region to its host galaxy’s total $\rm H\alpha$ light against the local broad-
band surface brightness in the immediate vicinity of that $\rm H\text{\small
II}$ region (excluding the region itself). We found that the $\rm
H\text{\small II}$ regions in UGC 8839 fall on the upper right envelope of the
distribution; for their luminosities, they are, on average, farther out in the
disc of UGC 8839 than comparable regions in our other sample galaxies.

Although this investigation was motivated by the striking appearance of the
ABC complex, we find that the entire population of $\rm H\text{\small II}$
regions in UGC 8839 is unusual. Given that the ABC complex does not, in fact,
stand out as unusual when taken in the context of the smaller $\rm
H\text{\small II}$ regions in UGC 8839, we conclude there is no compelling
evidence that the ABC complex is an intruder object, and that it can be fully
explained within the context of the $\rm H\text{\small II}$ region population
in UGC 8839.

###  5.3 UGC 8839 as a Mini-Malin 1

Alternatively, if indeed UGC 8839 has experienced a recent merger with a
smaller gas-rich satellite, the intruder may have been absorbed or disrupted,
and its gas is now fuelling star formation within the disc of UGC 8839. If
this kind of event is responsible for most or all of the $\rm H\text{\small
II}$ regions in UGC 8839, then we would not expect the ABC complex to stand
out as distinct.

This hypothesis has precedent: In the last decade a number of works have shown
that gas-rich mergers are able to stimulate starbursts in small galaxies just
as they are in more massive galaxies. As a somewhat extreme but well-studied
example, Ekta et al. (2008) report a disturbed $\rm H\text{\small I}$
morphology in the blue compact dwarf galaxy DDO 68, suggesting that the
current star formation is merger driven; this was followed by several detailed
studies of the stellar population and ISM metallicity (Martínez-Delgado et
al., 2012; Annibali et al., 2016), which confirm that DD0 68 is a late-stage
merger between the main progenitor of DDO 68 and a smaller gas-rich satellite.
Although DDO 68 is much smaller than UGC 8839 ($M_{g}=-16.2$ mag. versus
$M_{g}=-17.3$ mag.), they do have similar gas-mass to light ratios:
$\mathcal{M}_{\rm HI}/L_{g}=2.4$ for DDO 68 (Ekta et al., 2008) and
$\mathcal{M}_{\rm HI}/L_{g}=2.6$ for UGC 8838 (Richards et al., 2016) (solar
units).

However, the $\rm H\text{\small I}$ map of UGC 8839 shown in Figure A15 of
Richards et al. (2016) shows that, although the $\rm H\text{\small I}$ disc
does extend out as far as the ABC complex, it shows no signs of disrupted
morphology, unlike the disturbed $\rm H\text{\small I}$ morphology in DDO 68.

Additionally, the $\rm H\text{\small I}$ gas and the star formation in DDO 68
are located near the galactic centre (Pustilnik et al., 2005; Ekta et al.,
2008), consistent with the angular moment loss associated with a typical
accretion event. In practice, it would be difficult to arrange the merger and
disruption of a gas-rich dwarf without a significant loss of angular momentum
and disrupted morphology, which is not seen in UGC 8839.

A three-body interaction may be the solution to this puzzle. The giant LSB
galaxy Malin 1, originally thought to be an early-type galaxy, is now known to
have a very faint, blue, gas-rich disc with an incredible scale length of
$\sim$200kpc. The origin of the faint blue disc remained a puzzle until Zhu et
al. (2018) reported a Malin 1 analogue in the IllustrisTNG simulation. Looking
backward in the simulation, they find that the cause of the extended gas disc
is a three-body interaction, wherein an interacting pair of gas-rich galaxies
merge with a larger gas-poor object. This three-body interaction allows the
larger galaxy to acquire the satellites’ gas without significantly heating it
or driving it to the centre of the newly merged core, forming a diffuse
extended disk. It may be that UGC 8839 is a miniature analogue to Malin 1.

In this scenario, a three-body interaction delivered gas to UGC 8839, creating
the outer disc, and the merging satellites have since been absorbed or
disrupted. As with the proposed origin for Malin 1, the particular details of
the three-body interaction would lead to the gas being removed from the
satellites while retaining enough angular momentum to form the extended gas
disc of UGC 8839. Our findings here, that star formation in UGC 8839 primarily
occurs in the extreme outer disk, are consistent with this scenario.

###  5.4 UGC 8839 as a Hyper-Stable Disc

Although a three-body interaction may explain the origin of the extended disc
and the ABC complex, further explanation is required for UGC 8839 as a whole.
The $\rm H\text{\small I}$ maps reported in Richards et al. (2016) show that
the $\rm H\text{\small I}$ is more concentrated near the centre of UGC 8839
than near the ABC complex. Although the ABC complex is located at a local $\rm
H\text{\small I}$ density maximum, that maximum is less dense than the $\rm
H\text{\small I}$ at the centre of the galaxy where we see very little $\rm
H\alpha$ emission. The most curious aspect of UGC 8839 is not that it supports
star formation, but rather that the star formation does not seem to follow the
gas.

A plausible explanation is put forth in Garg & Banerjee (2017), wherein the
authors calculate the modified Toomre parameter developed in Romeo & Wiegert
(2011) for a sample of LSB discs. This modified Toomre parameter accounts for
gas as well as stars; since LSB discs are typically more gas rich than HSB
discs, they consider the possibility that too much gas may ‘over stabilise’ a
disc against star formation. However, Garg & Banerjee (2017) find that the
most significant impact on LSB disc stability comes from the fact they tend to
be dark matter dominated at all radii (de Blok & McGaugh, 1997; Simon et al.,
2003; Banerjee et al., 2010; Kuzio de Naray & Spekkens, 2011). As a result,
several of the galaxies in their sample reach stability minima only in the
outer disc, several scale lengths from the centre.

The dynamical analysis based on the $\rm H\text{\small I}$ velocity field data
presented in Richards et al. (2016) confirms that, for reasonable assumptions
about baryonic $\mathcal{M}/L$, UGC 8839 is dark matter dominated all the way
to its centre. Indeed, UGC 8839 is overall the most dark matter dominated
galaxy in the sample of 15 galaxies presented in Richards et al. (2016).

We find this to be the most likely interpretation, that a higher dark matter
fraction and greater dark matter dominance causes the disc of UGC 8839 to be
more stable than the other galaxies in our sample. This causes the local
stability minimum to be located beyond the edge of the visible disc, and
explains the preference of star formation for the outer disc of seen in Figure
5. The ABC complex is explained as a structure native to UGC 8839, which is
consistent with the statistical analysis of the HRLF presented in Section 3.
The extreme edge-dominated nature of star formation in UGC 8839, of which the
ABC complex is simply the most visible example, is indicative of galactic
dynamics dominated by dark matter at all radii.

##  6 Summary

We examined new $\rm H\alpha$ images and archival DESI Legacy Imaging Survey
g,r,z-band images of four LSB spirals and one HSB spiral. We used this
information to assess the significance of the large $\rm H\text{\small II}$
region complex in the extreme outer disc of the LSB spiral UGC 8839 (the ABC
complex), and then used this information to discuss the origins and nature of
the ABC complex.

Adopting a power law $\rm H\text{\small II}$ region luminosity function, we
determined the probability density function for the power law index for each
galaxy. Using these probability density curves, we then calculated the
likelihood that each galaxy would host its most luminous $\rm H\text{\small
II}$ regions given a random sample equal to the number of actual regions in
that galaxy. We find that the likelihood of the ABC complex being present in
UGC 8839 given the distribution of smaller regions is moderate, 40%-50%, and
conclude that the presence of the ABC complex in UGC 8839 is not extraordinary
since it can be fully accounted as part of the ensemble of $\rm H\text{\small
II}$ regions.

We then examined the spatial distribution of $\rm H\text{\small II}$ regions
in all of our sample galaxies, and find that the $\rm H\text{\small II}$
regions in UGC 8839 are preferentially found multiple scale lengths from the
centre, whereas the other galaxies in our sample have a more centrally
concentrated distribution. We conclude that the spatial distribution of $\rm
H\text{\small II}$ regions in UGC 8839 is unusual, and that the ABC complex is
simply the most visible example.

The most likely interpretation of these findings is that the disc of UGC 8839
is extremely stable against cloud formation because of a high dark matter
fraction (Richards et al., 2016). In particular, UGC 8839 is likely dark
matter dominated even in its central regions, which would raise the Toomre and
modified Toomre parameters ($Q$,$Q_{RW}$) to levels that inhibit star
formation and push the stability minimum past the edge of the visible galactic
disc.

This highlights the fundamental link between the dark matter halos and star-
formation histories of galaxies. UGC 8839 is a fairly extreme example in both
categories, and hints at the possibility that galaxies with greater
dark/baryonic matter fractions might show an even greater proclivity for edge-
dominated star formation.

A three-body interaction, similar to the interaction suspected to be
responsible for the extended gas disc in the giant LSB galaxy Malin 1 (Zhu et
al., 2018), may be able to explain the extended gas disc in UGC 8839. However,
a three-body interaction would not, by itself, be able to account for the
anti-correlation of $\rm H\text{\small I}$ density and star formation, wherein
the core of UGC 8839 has a high $\rm H\text{\small I}$ density but very little
star formation.

## Acknowledgements

We gratefully acknowledge the financial support of Five Colleges, Inc. and the
technical support of the staff of the WIYN 0.9m telescope at KPNO.

Based on observations at Kitt Peak National Observatory, NSF’s National
Optical-Infrared Astronomy Research Laboratory, which is operated by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

The Legacy Surveys consist of three individual and complementary projects: the
Dark Energy Camera Legacy Survey (DECaLS; NOAO Proposal ID # 2014B-0404; PIs:
David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO
Proposal ID # 2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band
Legacy Survey (MzLS; NOAO Proposal ID # 2016A-0453; PI: Arjun Dey). DECaLS,
BASS and MzLS together include data obtained, respectively, at the Blanco
telescope, Cerro Tololo Inter-American Observatory, National Optical Astronomy
Observatory (NOAO); the Bok telescope, Steward Observatory, University of
Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOAO. The
Legacy Surveys project is honored to be permitted to conduct astronomical
research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance
to the Tohono O’odham Nation.

NOAO is operated by the Association of Universities for Research in Astronomy
(AURA) under a cooperative agreement with the National Science Foundation.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de
Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia,
Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energeticas, Medioambientales y Tecnologicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de
Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-
Maximilians Universitat Munchen and the associated Excellence Cluster
Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, the Ohio State University, the
University of Pennsylvania, the University of Portsmouth, SLAC National
Accelerator Laboratory, Stanford University, the University of Sussex, and
Texas A&M University.

BASS is a key project of the Telescope Access Program (TAP), which has been
funded by the National Astronomical Observatories of China, the Chinese
Academy of Sciences (the Strategic Priority Research Program “The Emergence of
Cosmological Structures” Grant # XDB09000000), and the Special Fund for
Astronomy from the Ministry of Finance. The BASS is also supported by the
External Cooperation Program of Chinese Academy of Sciences (Grant #
114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #
11433005).

The Legacy Survey team makes use of data products from the Near-Earth Object
Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet
Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by
the National Aeronautics and Space Administration.

The Legacy Surveys imaging of the DESI footprint is supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility under the
same contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOAO.

## Data Availability

The $\rm H\alpha$ data underlying this article will be shared on reasonable
request to the corresponding author, and are also available via the NOAO
Science Archive:

http://archive1.dm.noao.edu/search/query/

The SDSS data are publicly available via the SDSS sky server:

http://skyserver.sdss.org/dr16/en/tools/chart/navi.aspx

## References

  * Ahumada et al. (2020) Ahumada R., et al., 2020, ApJS, 249, 3
  * Annibali et al. (2016) Annibali F., et al., 2016, ApJ, 826, L27
  * Banerjee et al. (2010) Banerjee A., Matthews L. D., Jog C. J., 2010, New Astron., 15, 89
  * Boissier et al. (2008) Boissier S., et al., 2008, ApJ, 681, 244
  * Bresolin & Kennicutt (2015) Bresolin F., Kennicutt R. C., 2015, MNRAS, 454, 3664
  * Caldwell et al. (1991) Caldwell N., Kennicutt R., Phillips A. C., Schommer R. A., 1991, ApJ, 370, 526
  * Cardelli et al. (1989) Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ, 345, 245
  * Ekta et al. (2008) Ekta Chengalur J. N., Pustilnik S. A., 2008, MNRAS, 391, 881
  * Garg & Banerjee (2017) Garg P., Banerjee A., 2017, MNRAS, 472, 166
  * Helmboldt et al. (2009) Helmboldt J. F., Walterbos R. A. M., Bothun G. D., O’Neil K., Oey M. S., 2009, MNRAS, 393, 478
  * Huang et al. (2013) Huang M.-L., Kauffmann G., Chen Y.-M., Moran S. M., Heckman T. M., Davé R., Johansson J., 2013, MNRAS, 431, 2622
  * Kennicutt et al. (1989) Kennicutt Jr. R. C., Edgar B. K., Hodge P. W., 1989, ApJ, 337, 761
  * Kim (2007) Kim J. H., 2007, PhD thesis, University of Maryland, College Park
  * Kuzio de Naray & Spekkens (2011) Kuzio de Naray R., Spekkens K., 2011, ApJ, 741, L29
  * Kuzio de Naray et al. (2004) Kuzio de Naray R., McGaugh S. S., de Blok W. J. G., 2004, MNRAS, 355, 887
  * Martínez-Delgado et al. (2012) Martínez-Delgado D., et al., 2012, ApJ, 748, L24
  * McCall et al. (1985) McCall M. L., Rybski P. M., Shields G. A., 1985, ApJS, 57, 1
  * McGaugh et al. (1995) McGaugh S. S., Schombert J. M., Bothun G. D., 1995, AJ, 109, 2019
  * O’Neil et al. (2007) O’Neil K., Oey M. S., Bothun G., 2007, AJ, 134, 547
  * Oey & Kennicutt (1993) Oey M. S., Kennicutt Jr. R. C., 1993, ApJ, 411, 137
  * Pustilnik et al. (2005) Pustilnik S. A., Kniazev A. Y., Pramskij A. G., 2005, A&A, 443, 91
  * Richards et al. (2016) Richards E. E., et al., 2016, MNRAS, 460, 689
  * Romeo & Wiegert (2011) Romeo A. B., Wiegert J., 2011, MNRAS, 416, 1191
  * Schlegel et al. (1998) Schlegel D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525
  * Simon et al. (2003) Simon J. D., Bolatto A. D., Leroy A., Blitz L., 2003, ApJ, 596, 957
  * Vorobyov et al. (2009) Vorobyov E. I., Shchekinov Y., Bizyaev D., Bomans D., Dettmar R.-J., 2009, A&A, 505, 483
  * Young et al. (2015) Young J. E., Kuzio de Naray R., Wang S. X., 2015, MNRAS, 452, 2973
  * Young et al. (2020) Young J. E., Kuzio de Naray R., Wang S. X., 2020, MNRAS, 493, 55
  * Zhu et al. (2018) Zhu Q., et al., 2018, MNRAS, 480, L18
  * de Blok & McGaugh (1997) de Blok W. J. G., McGaugh S. S., 1997, MNRAS, 290, 533
  * de Blok & van der Hulst (1998) de Blok W. J. G., van der Hulst J. M., 1998, A&A, 335, 421
  * de Blok et al. (1995) de Blok W. J. G., van der Hulst J. M., Bothun G. D., 1995, MNRAS, 274, 235
  * de Blok et al. (1996) de Blok W. J. G., McGaugh S. S., van der Hulst J. M., 1996, MNRAS, 283, 18
  * van den Hoek et al. (2000) van den Hoek L. B., de Blok W. J. G., van der Hulst J. M., de Jong T., 2000, A&A, 357, 397

##  Appendix A Data on Individual Regions

Table 3: Properties of Individual $\rm H\text{\small II}$ Regions | $f_{\rm H\alpha}{}$ | $\log{L_{\rm H\alpha}{}}$ | $\Delta\log{L_{\rm H\alpha}{}}$ | $\frac{f_{\rm H\alpha}{}}{f_{\rm H\alpha}{}(tot)}$ | $\mu(g)$ | $\Delta\mu(g)$ | $\mu(r)$ | $\Delta\mu(r)$ | $\mu(z)$ | $\Delta\mu(z)$
---|---|---|---|---|---|---|---|---|---|---
| $\rm 10^{-15}erg\,s^{-1}\,cm^{-2}$ | $\log{\rm erg\,s^{-1}}$ |  |  | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ | $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$
UGC8839 |  |  |  |  |  |  |  |  |  |
A | 6.81 | 38.60 | 0.01 | 0.117 | 27.0 | 0.13 | 26.9 | 0.20 | 27.6 | 1.06
B | 8.01 | 38.67 | 0.01 | 0.138 | 26.5 | 0.08 | 26.4 | 0.13 | 26.2 | 0.28
C | 2.40 | 38.15 | 0.04 | 0.041 | 26.3 | 0.09 | 26.1 | 0.11 | 26.5 | 0.41
D | 1.79 | 38.02 | 0.08 | 0.031 | 28.9 | 0.41 | 28.3 | 0.40 | — | —
E | 2.37 | 38.15 | 0.04 | 0.041 | 25.4 | 0.04 | 25.2 | 0.06 | 24.9 | 0.11
F | 2.49 | 38.17 | 0.04 | 0.043 | 27.5 | 0.30 | 27.1 | 0.33 | — | —
G | 1.68 | 38.00 | 0.05 | 0.029 | 26.3 | 0.09 | 26.1 | 0.14 | 26.7 | 0.63
H | 0.98 | 37.76 | 0.07 | 0.017 | 26.7 | 0.12 | 26.5 | 0.21 | 26.2 | 0.34
I | 0.28 | 37.26 | 0.19 | 0.005 | 26.8 | 0.18 | 26.4 | 0.24 | 26.0 | 0.44
J | 0.38 | 37.42 | 0.17 | 0.006 | — | 15.64 | — | — | — | —
K | 1.33 | 38.00 | 0.10 | 0.023 | 23.3 | 0.01 | 22.9 | 0.01 | 22.7 | 0.01
L | 0.73 | 37.78 | 0.13 | 0.012 | 26.7 | 0.16 | 26.8 | 0.25 | — | —
M | 0.40 | 37.55 | 0.23 | 0.007 | 26.7 | 0.15 | 26.9 | 0.29 | 31.5 | 47.57
N | 0.45 | 37.60 | 0.16 | 0.008 | 23.2 | 0.01 | 22.9 | 0.01 | 22.6 | 0.03
O | 1.03 | 37.96 | 0.09 | 0.018 | 26.7 | 0.06 | 26.5 | 0.10 | 27.3 | 0.44
P | 0.63 | 37.75 | 0.11 | 0.011 | 23.7 | 0.03 | 23.3 | 0.02 | 23.1 | 0.03
Q | 0.73 | 37.81 | 0.09 | 0.013 | 24.8 | 0.04 | 24.6 | 0.04 | 24.3 | 0.07
Total | 58.13 | 39.54 | 0.07 | 1.000 |  |  |  |  |  |
Diffuse | 28.50 | 39.23 | 0.14 | 0.490 |  |  |  |  |  |
NGC4455 |  |  |  |  |  |  |  |  |  |
A | 41.46 | 38.73 | 0.01 | 0.083 | 23.1 | 0.03 | 22.9 | 0.03 | 22.9 | 0.04
B | 68.44 | 38.95 | 0.01 | 0.137 | 21.6 | 0.01 | 21.3 | 0.01 | 21.2 | 0.01
C | 11.83 | 38.19 | 0.03 | 0.024 | 22.7 | 0.02 | 22.4 | 0.02 | 22.4 | 0.02
D | 42.57 | 38.74 | 0.02 | 0.085 | 21.6 | 0.02 | 21.2 | 0.02 | 21.0 | 0.02
E | 4.54 | 37.77 | 0.08 | 0.009 | 21.0 | 0.01 | 20.6 | 0.01 | 20.4 | 0.01
F | 6.41 | 37.92 | 0.05 | 0.013 | 21.2 | 0.02 | 20.8 | 0.02 | 20.6 | 0.02
G | 3.23 | 37.62 | 0.09 | 0.006 | 22.3 | 0.02 | 21.9 | 0.02 | 21.7 | 0.02
H | 24.08 | 38.49 | 0.02 | 0.048 | 22.2 | 0.03 | 21.8 | 0.02 | 21.6 | 0.02
I | 14.77 | 38.28 | 0.03 | 0.029 | 21.8 | 0.01 | 21.5 | 0.01 | 21.3 | 0.01
J | 2.80 | 37.56 | 0.09 | 0.006 | 21.8 | 0.02 | 21.5 | 0.02 | 21.4 | 0.02
K | 2.34 | 37.48 | 0.12 | 0.005 | 22.4 | 0.04 | 22.1 | 0.03 | 21.9 | 0.03
L | 6.83 | 37.95 | 0.04 | 0.014 | 22.5 | 0.01 | 22.2 | 0.01 | 22.0 | 0.01
M | 2.62 | 37.53 | 0.09 | 0.005 | 22.1 | 0.03 | 21.8 | 0.03 | 21.7 | 0.03
N | 17.06 | 38.34 | 0.03 | 0.034 | 23.1 | 0.02 | 22.8 | 0.02 | 22.6 | 0.02
O | 5.27 | 37.83 | 0.06 | 0.011 | 22.5 | 0.02 | 22.2 | 0.02 | 21.9 | 0.03
P | 9.86 | 38.11 | 0.04 | 0.020 | 22.2 | 0.02 | 21.8 | 0.02 | 21.6 | 0.02
Q | 25.03 | 38.59 | 0.02 | 0.050 | 21.3 | 0.01 | 20.9 | 0.01 | 20.7 | 0.01
R | 4.81 | 37.95 | 0.08 | 0.010 | 22.2 | 0.01 | 21.8 | 0.01 | 21.6 | 0.01
S | 11.32 | 38.39 | 0.05 | 0.023 | 23.5 | 0.03 | 23.2 | 0.02 | 23.2 | 0.04
T | 4.23 | 38.02 | 0.08 | 0.008 | 22.3 | 0.02 | 22.0 | 0.02 | 21.8 | 0.02
U | 4.39 | 38.10 | 0.08 | 0.009 | 21.6 | 0.02 | 21.2 | 0.02 | 21.0 | 0.02
V | 3.00 | 37.99 | 0.10 | 0.006 | 22.3 | 0.01 | 22.0 | 0.01 | 21.8 | 0.02
W | 2.94 | 38.03 | 0.10 | 0.006 | 23.0 | 0.03 | 22.5 | 0.03 | 22.3 | 0.03
X | 3.61 | 38.16 | 0.07 | 0.007 | 22.5 | 0.03 | 22.4 | 0.03 | 22.4 | 0.04
Y | 4.62 | 38.32 | 0.10 | 0.009 | 23.1 | 0.01 | 22.8 | 0.01 | 22.6 | 0.02
Total | 500.78 | 39.81 | 0.02 | 1.000 |  |  |  |  |  |
Diffuse | 172.72 | 39.35 | 0.05 | 0.345 |  |  |  |  |  |
Table 4: continued

Properties of Individual $\rm H\text{\small II}$ Regions  $f_{\rm H\alpha}{}$
$\log{L_{\rm H\alpha}{}}$ $\Delta\log{L_{\rm H\alpha}{}}$ $\frac{f_{\rm
H\alpha}{}}{f_{\rm H\alpha}{}(tot)}$ $\mu(g)$ $\Delta\mu(g)$ $\mu(r)$
$\Delta\mu(r)$ $\mu(z)$ $\Delta\mu(z)$ $\rm 10^{-15}erg\,s^{-1}\,cm^{-2}$
$\log{\rm erg\,s^{-1}}$ $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$
$\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$
UGC6181 A 0.53 37.55 0.06 0.002 24.8 0.03 24.4 0.03 23.8 0.04 B 6.67 38.65
0.01 0.029 24.2 0.02 23.7 0.02 23.4 0.02 C 0.49 37.52 0.10 0.002 25.4 0.04
25.1 0.05 24.7 0.06 D 6.09 38.61 0.01 0.026 23.8 0.02 23.4 0.02 23.2 0.02 E
1.21 37.91 0.04 0.005 24.5 0.02 24.1 0.03 23.9 0.04 F 1.23 37.91 0.04 0.005
23.1 0.02 22.7 0.02 22.5 0.02 G 2.09 38.15 0.02 0.009 22.8 0.03 22.4 0.02 22.1
0.02 H 0.73 37.69 0.05 0.003 22.4 0.02 22.1 0.02 21.8 0.02 I 0.42 37.45 0.10
0.002 23.3 0.01 23.0 0.02 22.8 0.02 J 0.08 36.72 0.34 0.000 24.6 0.03 24.4
0.04 24.2 0.04 K 0.20 37.12 0.15 0.001 22.3 0.01 21.9 0.01 21.6 0.01 L 0.23
37.18 0.15 0.001 22.3 0.01 21.9 0.01 21.6 0.01 M 1.12 37.88 0.05 0.005 22.7
0.02 22.3 0.01 22.0 0.01 N 0.92 37.79 0.04 0.004 23.4 0.04 22.9 0.03 22.4 0.03
O 2.08 38.14 0.02 0.009 23.4 0.03 23.0 0.03 22.6 0.02 P 3.51 38.37 0.02 0.015
22.8 0.01 22.4 0.01 22.1 0.01 Q 0.60 37.61 0.07 0.003 24.7 0.03 24.2 0.03 24.0
0.04 R 0.36 37.38 0.08 0.002 22.8 0.01 22.3 0.01 22.1 0.01 S 0.98 37.82 0.04
0.004 22.5 0.01 22.0 0.01 21.7 0.02 T 1.88 38.10 0.03 0.008 22.8 0.02 22.3
0.02 22.0 0.02 U 1.12 37.87 0.04 0.005 23.2 0.01 22.8 0.01 22.5 0.02 V 1.03
37.84 0.04 0.004 23.0 0.02 22.6 0.02 22.3 0.02 W 0.44 37.47 0.09 0.002 22.9
0.01 22.5 0.01 22.2 0.01 X 0.90 37.78 0.05 0.004 23.2 0.02 22.8 0.02 22.5 0.02
Y 4.70 38.50 0.01 0.020 23.7 0.03 23.3 0.03 23.0 0.03 Z 2.25 38.18 0.02 0.010
23.4 0.02 23.0 0.02 22.7 0.02 AA 1.21 37.91 0.03 0.005 25.3 0.04 24.7 0.04
24.1 0.04 AB 0.86 37.76 0.05 0.004 23.3 0.06 23.0 0.05 22.7 0.04 AC 0.10 36.85
0.31 0.000 23.5 0.02 23.0 0.02 22.8 0.02 Total 233.27 40.19 0.01 1.000 Diffuse
189.25 40.10 0.02 0.811 UGC6151 A 0.90 37.80 0.05 0.017 24.9 0.02 24.6 0.03
24.5 0.06 B 0.54 37.58 0.10 0.010 25.9 0.04 25.9 0.08 25.6 0.12 C 0.87 37.79
0.03 0.017 24.3 0.03 24.0 0.03 23.8 0.05 D 1.42 38.00 0.02 0.027 24.2 0.02
23.8 0.02 23.6 0.04 E 0.79 37.75 0.07 0.015 25.1 0.03 24.8 0.04 24.7 0.07 F
0.79 37.74 0.06 0.015 23.7 0.01 23.3 0.01 23.0 0.01 G 0.24 37.23 0.14 0.004
24.6 0.02 24.2 0.02 24.0 0.03 H 2.19 38.19 0.03 0.041 24.4 0.01 24.2 0.02 23.8
0.02 I 0.71 37.70 0.08 0.013 25.1 0.03 24.9 0.04 24.7 0.07 J 0.96 37.83 0.06
0.018 24.2 0.01 23.8 0.02 23.6 0.03 K 0.59 37.62 0.07 0.011 23.6 0.01 23.1
0.01 22.8 0.01 L 0.72 37.70 0.06 0.014 23.1 0.01 22.6 0.01 22.3 0.01 M 0.47
37.52 0.07 0.009 23.3 0.01 22.8 0.01 22.5 0.01 N 8.79 38.79 0.01 0.166 24.0
0.01 23.6 0.01 23.3 0.02 O 2.04 38.16 0.03 0.039 24.7 0.01 24.4 0.02 24.1 0.03
P 1.04 37.87 0.06 0.020 25.0 0.02 24.7 0.03 24.6 0.05 Q 0.87 37.79 0.03 0.016
25.0 0.03 24.9 0.04 24.9 0.09 R 0.40 37.45 0.07 0.008 25.1 0.03 25.0 0.05 24.7
0.07 S 3.53 38.40 0.02 0.067 24.7 0.02 24.4 0.02 24.3 0.04 T 4.08 38.46 0.01
0.077 25.0 0.02 24.7 0.03 24.6 0.06 U 3.42 38.38 0.02 0.065 24.7 0.03 24.3
0.04 24.2 0.05 V 0.92 37.81 0.03 0.017 23.9 0.02 23.5 0.02 23.2 0.03 W 0.54
37.58 0.08 0.010 23.4 0.02 23.0 0.07 22.6 0.11 X 0.50 37.55 0.08 0.009 23.2
0.01 22.8 0.01 22.5 0.01 Y 1.55 38.04 0.02 0.029 25.0 0.02 24.8 0.04 24.8 0.08
Total 52.90 39.57 0.02 1.000 Diffuse 14.06 39.00 0.08 0.266

Table 5: continued

Properties of Individual $\rm H\text{\small II}$ Regions  $f_{\rm H\alpha}{}$
$\log{L_{\rm H\alpha}{}}$ $\Delta\log{L_{\rm H\alpha}{}}$ $\frac{f_{\rm
H\alpha}{}}{f_{\rm H\alpha}{}(tot)}$ $\mu(g)$ $\Delta\mu(g)$ $\mu(r)$
$\Delta\mu(r)$ $\mu(z)$ $\Delta\mu(z)$ $\rm 10^{-15}erg\,s^{-1}\,cm^{-2}$
$\log{\rm erg\,s^{-1}}$ $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$
$\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm
mag/{{}_{\Box^{{}^{\prime\prime}}}}$ $\rm mag/{{}_{\Box^{{}^{\prime\prime}}}}$
UGC5633 A 1.12 37.87 0.08 0.008 29.4 1.38 22.9 0.02 22.6 0.02 B 0.59 37.58
0.10 0.004 28.9 0.88 — — — — C 0.50 37.51 0.11 0.004 25.0 0.03 24.5 0.03 24.3
0.06 D 0.76 37.70 0.10 0.005 25.1 0.03 24.7 0.04 24.7 0.10 E 1.38 37.96 0.07
0.010 25.3 0.04 24.8 0.05 24.5 0.08 F 1.21 37.90 0.07 0.009 24.5 0.02 24.1
0.02 24.1 0.05 G 0.59 37.59 0.10 0.004 23.6 0.01 23.2 0.01 23.0 0.02 H 4.11
38.43 0.02 0.029 23.4 0.02 22.9 0.02 22.6 0.03 I 3.97 38.42 0.03 0.028 23.2
0.02 22.7 0.02 22.5 0.02 J 3.29 38.33 0.04 0.024 23.7 0.02 23.3 0.02 23.0 0.02
K 1.26 37.92 0.08 0.009 25.2 0.03 24.6 0.04 24.4 0.09 L 0.54 37.55 0.13 0.004
25.8 0.06 25.4 0.08 25.4 0.19 M 0.99 37.81 0.09 0.007 26.7 0.12 26.6 0.20 27.1
0.72 N 2.26 38.17 0.04 0.016 24.3 0.01 23.9 0.02 23.8 0.04 O 2.16 38.15 0.05
0.015 24.0 0.02 23.6 0.02 23.4 0.03 P 5.61 38.57 0.03 0.040 23.6 0.02 23.0
0.02 22.7 0.02 Q 3.69 38.38 0.03 0.026 23.3 0.03 22.8 0.03 22.6 0.03 R 2.00
38.12 0.06 0.014 24.9 0.03 24.6 0.04 24.4 0.10 S 11.63 38.88 0.01 0.083 24.3
0.02 23.8 0.02 23.6 0.04 T 0.92 37.78 0.11 0.007 25.7 0.06 25.6 0.09 26.0 0.32
U 3.15 38.31 0.04 0.022 25.8 0.05 25.5 0.07 25.8 0.21 V 1.37 37.95 0.07 0.010
24.0 0.03 23.6 0.02 23.5 0.03 W 0.64 37.62 0.10 0.005 23.8 0.04 23.4 0.03 23.3
0.04 X 2.21 38.16 0.07 0.016 24.1 0.02 23.8 0.02 23.6 0.03 Y 0.51 37.52 0.15
0.004 24.6 0.04 24.2 0.03 24.2 0.06 Z 0.64 37.62 0.14 0.005 25.9 0.05 25.6
0.08 25.6 0.22 AA 1.75 38.10 0.09 0.013 23.4 0.02 23.3 0.01 23.0 0.02 AB 2.39
38.27 0.07 0.017 23.7 0.01 26.2 0.11 27.3 0.86 AC 0.92 37.89 0.12 0.007 26.1
0.06 23.5 0.01 23.2 0.02 AD 0.50 37.65 0.18 0.004 23.8 0.02 23.3 0.01 23.1
0.02 AE 0.72 37.84 0.18 0.005 23.7 0.02 24.6 0.05 24.0 0.08 AF 0.45 37.67 0.20
0.003 25.1 0.03 24.7 0.06 24.6 0.13 AG 0.78 37.94 0.11 0.006 25.0 0.04 — — — —
Bar 6.89 38.65 0.04 0.049 23.4 Total 139.86 39.96 0.02 1.000 23.7 Diffuse
75.24 39.69 0.04 0.538 26.1