We differentiate the regions and DIG based on the ionized gas surface brightness, but compare the equivalent width between these areas of emission. Interestingly we find no significant difference between the distribution of   equivalent width between spaxels identified as belonging to an region and that belonging to DIG in IC 10. This may be a selection effect of our study. The high density of regions not only gives a small sample of diffuse gas, but also makes it likely to be more closely associated with the inter-dispersed regions than is typical of DIG studies. However, the DIG in IC 10 may be ionized by different sources than typically observed. [Hidalgo-Gámez, 2005] finds higher excitation in the IC 10 DIG than for spiral galaxies which they find can be produced by leakage from regions and the large number of WR stars. The gas throughout IC 10, and particularly in the KCWI FoV, may be more similar to what Lacerda et al., 2018 refers to as mDIG, or 'mixed' DIG in which the ionization source is due to a combination of processes such as emission from an older stellar population in addition to photon leakage from regions.  This is consistent with the small difference in metallicity we see between the regions and surrounding gas in comparison with other studies, as well as Polles et al., 2019 Cloudy simulations showing matter-bounded regions in IC 10 which would result in escaping photons ionizing the DIG.

§.§ Scaling Relations

In Cosens et al., 2018 we developed a framework to use Bayesian inference via PyStan to fit the scaling relationships between the properties of local and high-redshift star-forming regions in the literature. In this study we focused primarily on the relationship between star forming region size and   luminosity (), which takes on the form of a power law:

\begin{equation}
L_{\rm H\alpha}=exp(\beta)r^{\alpha}_{\rm clump}
\end{equation}

with $r_{\rm clump}$ giving the radius of the star forming region and $\beta$ giving the intercept of the fit. The key model parameter of interest is the slope, $\alpha$. The value of this slope holds information about the driving formation mechanism of the star-forming regions. A slope of $r^3$ is often explained by a region which forms under Jeans collapse and is then well-represented by a   sphere. On the other hand, a slope of $r^2$ is often explained by a region which forms under Toomre instability and undergoes the fastest mode of Jeans collapse resulting in a different form for the characteristic mass and size <cit.>. In Cosens et al., 2018 we found that this $r^2$ slope could also be explained by a   sphere argument where the ionizing photon production rate is large enough that the radius of the region is larger than the scale height of the galaxy disk. This would lead to a non-spherical geometry and an observed relationship of $L\sim r^2$.

Using Bayesian inference provides a number of advantages over standard least-squares fitting. First, uncertainties in every dimension can be incorporated in the fitting; there is no need to estimate a single overall uncertainty assigned to one dimension. Second, we can use our existing knowledge of the scaling relationships between parameters to inform our model through the use of Bayesian priors. Third, this method reproduces a distribution for each model parameter, allowing the determination of not only the best fit model, but robust determination of uncertainties for each model component as well. Using this framework allowed us to perform robust fits to the overall scaling relationships as well as investigate potential differences in smaller subsamples such as redshift bins and lensed versus field galaxies. Interestingly, we were able to identify a possible break in the size-luminosity scaling relationship based on the  of the star-forming region.

However, a key missing area of the parameter space in our previous investigation was small, low $\rm L_{H\alpha}$ star-forming regions ($<50$ pc, $<10^{34-36}$ ). This sets the limit on the low-mass end of the relationship and helps to constrain the intercept when performing fits. One challenge in interpreting fitting results for sub samples of the local and high-redshift clumps was that the best fit slope and intercept are not entirely independent parameters. Therefore, missing constraints on the low-mass intercept of the size-luminosity relationship makes it difficult to be certain whether a change in slope is really a change in that parameter or just in the lever arm of the fit. With the proximity of IC 10 and the sensitivity of KCWI we are able to target a large sample of regions at this crucial scale.

§.§.§ Size-Luminosity Relationship

While these small, low-mass star-forming regions are critical for constraining the intercept of the size-luminosity relationship, one must first check that the possibility of stochastic sampling is not biasing the measured properties. The lower mass limit to avoid stochastic effects is typically determined to be $\sim 10^3$  <cit.>. Below these masses random sampling of the IMF can lead to deviations between the actual physical properties and those determined from photometric measurements. The average stellar mass distribution determined for the IC 10 regions from the measured 5007  flux is $\rm M_* \sim (4\pm14)\times10^2 \, M_{\odot}$, with 96% falling below the $10^3$  limit. Further, IC 10's regions are largely consistent with ionization predominantly from a single O or B type star which will be stochastic by nature. Hannon et al., 2019 found themselves faced with a similar dilemma studying a sample of $\sim$700 young star clusters with over 90% below the stochastic limit. They investigate a method of mitigating the impact of stochastic sampling by stacking the fluxes of individual clusters with similar properties. These composite clusters no longer fall into the regime of stochastic sampling, but Hannon et al., 2019 still find results consistent with the individual clusters. We apply a similar check to our sample, but rather than stacking the region spectra, we instead make use of the hierarchical structure determined from  which identifies the region complexes. Using the complexes results in a smaller sample than with individual regions, but one that lies above the stochastic limit with a mean $\sim 2\times10^3$. We do not find any significant deviation in the trends determined for the complexes vs. the individual regions, finding proportional increases in the mass and luminosity to the increase in radius (e.g., Figure <ref>). The measured properties of individual regions may show an increase in scatter due to stochastic sampling, but the average sample properties and trends do not appear to be affected. Therefore we will proceed with the determination of the scaling relationships using the individual regions in the fitting, but we will include the complexes in all figures for comparison.

Fitting the size-luminosity relationship using our MCMC framework with just IC 10 regions yields a slope of $\rm L_{H\alpha} \propto r_{clump}^{3.6}$. The complexes do shift to slightly larger radii than the individual regions but with a proportional increase in luminosity, placing them along the same relationship. We also combine these regions with the full sample of regions and high-redshift clumps outlined in Cosens et al., 2018, now including additional published samples of star-forming regions in the SMC [Kennicutt & Hodge, 1986], LMC [Ambrocio-Cruz et al., 2016], NGC6822 [Hodge et al., 1989], and local LIRGs [Zaragoza-Cardiel et al., 2017, Larson et al., 2020]. These LIRG studies also make use of  to identify star-forming regions and their properties[The effective radii, $\rm r_{eff}$, of Zaragoza-Cardiel] et al., 2017 are adjusted to match our definition of $r_{1/2}^*$ for a consistent comparison.. Collating all low and high-redshift samples results in a significantly shallower slope of approximately $\propto r^3$; matching the result of fitting the full sample in Cosens et al., 2018 and indicating the IC 10 regions may be an outlier. The IC 10 fit and entire sample fit are shown in Figure <ref>, where it can be can see that the IC 10 regions lie above the size-luminosity relation found for the full sample. This is also true of some other local samples, particularly in more extreme environments such as the LIRGs <cit.> and turbulent galaxies [Fisher et al., 2017]. Whether the offset of the IC 10 regions is then due to improved resolution breaking the IC 10 regions down into more compact components, or due to a fundamental difference in the environments of star-forming regions driving scatter in the scaling relationships is not fully clear. It may be that active expansion of these regions as discussed in Section <ref> leads to the offset we see here with the regions currently being under-sized for their luminosity.

(a): Results of fitting the size-luminosity scaling relationship for IC 10 regions identified in our KCWI observations. (b): Results of fitting the size-luminosity scaling relationship for IC 10 regions (red stars) as well as the rest of the local and high-redshift sample. The IC 10 region complexes are shown for comparison (cyan stars) but are not included in the determination of the fit.

The full results of these fits and those described in Section <ref> are shown in Table <ref> for the intercept, slope, and intrinsic scatter along with their uncertainties. Results of fitting additional data subsets (as detailed in column 1) are also included for completeness.

Size - Luminosity Relation Fit Parameters: ($L_{H\alpha} = e^{\beta} r_{clump}^{\alpha}$)
Sample     Subset Criteria     Figure     $\alpha$     $\beta$     Scatter (r)     Scatter (L)     # of Clumps
IC 10         <ref>     $3.601^{+0.471}_{-0.385}$     $78.812^{+0.455}_{-0.652}$     $0.147^{+0.154}_{-0.103}$     $0.156^{+0.144}_{-0.103}$     46

312emall comparisons             $3.215^{+0.023}_{-0.030}$     $73.782^{+0.129}_{-0.119}$     $0.184^{+0.118}_{-0.112}$     $0.171^{+0.115}_{-0.114}$     3858

high         $1.788^{+0.063}_{-0.059}$     $84.617^{+0.314}_{-0.357}$     $0.461^{+0.317}_{-0.316}$     $0.470^{+0.320}_{-0.314}$     264

low         $2.967^{+0.024}_{-0.027}$     $74.670^{+0.123}_{-0.111}$     $0.142^{+0.092}_{-0.095}$     $0.136^{+0.092}_{-0.093}$     3573

310emIC 10 & all comparisons         <ref>     $3.002^{+0.023}_{-0.024}$     $74.800^{+0.100}_{-0.115}$     $0.240^{+0.147}_{-0.154}$     $0.249^{+0.154}_{-0.151}$     3904

high     <ref>     $1.841^{+0.070}_{-0.059}$     $84.310^{+0.335}_{-0.384}$     $0.426^{+0.279}_{-0.285}$     $0.360^{+0.323}_{-0.268}$     266

low     <ref>     $2.775^{+0.021}_{-0.018}$     $75.577^{+0.086}_{-0.090}$     $0.194^{+0.136}_{-0.132}$     $0.194^{+0.132}_{-0.138}$     3617

Results of model parameters determined from MCMC fitting of IC 10 regions and the local and high-redshift comparison sample for all fits discussed in Section <ref> as well as additional fits to only the comparison sample (rows 2-4).

§.§.§   break

In Cosens et al., 2018, we found that there was a potential break in the size-luminosity relationship that divides star-forming regions into two samples: one with high  and one with low   with the break nominally located at a value of $\rm\Sigma_{SFR}= 1 \, $. Both locally and at high redshift we found that lower   star-forming regions followed a size-luminosity relationship of  $L\sim r^3$, while the high   sub-sample followed a relationship closer to $L\sim r^2$.

In our observations of IC 10's regions, the average size of identified regions is $\rm r_{avg}=4.0 \, pc$ with a $\rm \Sigma_{SFR,avg}=0.20 \,$  with only 2 identified regions falling above the $\rm\Sigma_{SFR}> 1 \,$  limit. When combined with the full comparison sample this does still provide some improvements on constraining the size-luminosity relationship at low masses by reducing the uncertainties on the fitted parameters. The resulting slopes for the two populations remain consistent with the previous results, with nominal values differing by $\rm <2\sigma$.

(a): Size vs.   for regions identified in our observations of IC 10 as well as each comparison sample. These data are divided into high and low   based on a break at $\rm \Sigma_{SFR} = 1 \, $(dashed line). (b): Results of fitting the size-luminosity scaling relationship for these two groups. The low   subset tends towards a slop of $L\sim r^3$, while the high   subset produces a slope closer to $L\sim r^2$. The IC 10 region complexes are shown for comparison (cyan stars) but are not included in the determination of the fit.

§.§.§ Size - Velocity Dispersion

Larson, 1981 found an empirical relationship between the size of molecular clouds and their velocity dispersion that is tied to turbulence in the clouds and ISM. For resolved Milky Way GMCs in virial equilibrium, Larson, 1981 found a relationship:

\begin{equation}
\sigma = 1.1r^{0.38}
\end{equation}

where $r$ is the radius of the GMC in pc and $\sigma$ is in . This has been refined and updated with expanded galactic [Solomon et al., 1987] and extra-galactic [Bolatto et al., 2008] data sets of molecular clouds, arriving at slight changes to the scaling but the same basic conclusion that the line width increases with cloud size approximately $\rm \propto r^{1/2}$. This scaling is typically interpreted as being due to turbulence in the molecular clouds <cit.>.

While Larson's Law is usually discussed in relation to GMCs it may also apply to ionized gas in star-forming regions if they are in virial equilibrium and exhibit turbulent motions. Wisnioski et al., 2012 fit this relationship with a portion of the local and high-redshift data sets used in the present study and find a scaling of $\sigma \propto r^{0.42}$, but determine that their $\rm z\sim1$ clumps are likely not virialized. We test this now with the addition of local and high-z star-forming regions measured with IFS observations since the study by Wisnioski et al., 2012. Unfortunately, not all data sets used to investigate the size-luminosity relationship have spectral information, but it is still a large enough sample (515 regions) to investigate the presence of a similar size-velocity dispersion relationship. In fact, we do see a clear trend where the velocity dispersion increases with larger size regions, except with IC 10, which is still  a notable outlier. On their own, IC 10 regions do not show any evidence of this power law scaling, and instead when included significantly skew the resultant slope. Without IC 10 included we arrive at a relationship of $\sigma \propto r^{0.4}$ using the same MCMC framework as described in Section <ref>, in line with the results of Wisnioski et al., 2012 and similar to the GMC results. However, when the IC 10 regions are included, they clearly lie above the size - $\sigma$ relationship found for the other samples as shown in Figure <ref>. Furthermore, including the IC 10 regions in the fitting reduces the slope to $\sigma \propto r^{0.2}$ . This further supports the conclusion in Sections <ref> & <ref> that the regions identified in this study are not generally virialized. In fact, if we retain only the component of velocity dispersion due to rotational motion as identified in Section <ref>, then the identified regions in IC 10 follow the same relationship identified for the rest of the samples (see Figure <ref>b). There is a significant amount of scatter in the $\rm r - \sigma$ relationship, but it suggests that rotational motion is a much more significant source of the measured line widths in the other star-forming regions than in IC 10's regions.

Size vs. $\sigma$   relationship for the star-forming regions in each comparison sample containing spectral information. The fit is computed using our MCMC framework excluding the IC 10 regions (red stars) and complexes (cyan stars) from the analysis. The mean fit is shown with the green line while the uncertainty in the fit is denoted by the shaded green region. (a): Both the IC 10 regions and complexes fall above this mean fit, and in most cases above the uncertainty as well, indicating that these regions are not virialized. The complexes lie closer to the mean relationship while still tending towards higher $\sigma$ for a given size, further supporting the conclusion that the individual regions are not virialized as the increase in size does not result in a proportional increase in measured line width. (b): The IC 10 regions and complexes lie along the mean size - $\sigma$ after the excess velocity dispersion determined in Section <ref> is removed. The excess dispersion was found to be in addition to that produced by rotational motion of the region gas and is therefore likely due to outflows or expansion of the ionized gas.

Size - Velocity Dispersion Relation Fit Parameters: ($\sigma = \beta r_{clump}^{\alpha}$)
Sample     Figure     $\alpha$     $\beta$     Scatter (r)     Scatter ($\sigma$)     # of Clumps
all comparisons     <ref>b     $0.406^{+0.025}_{-0.025}$     $3.704^{+1.129}_{-1.154}$     $0.444^{+0.288}_{-0.276}$     $0.414^{+0.275}_{-0.294}$     469

IC 10 & all comparisons     <ref>a     $0.216^{+0.013}_{-0.012}$     $10.442^{+1.072}_{-1.060}$     $0.262^{+0.165}_{-0.171}$     $0.257^{+0.176}_{-0.172}$     515

Results of model parameters determined from MCMC fitting of IC 10 regions and the local and high-redshift comparison sample for the size-velocity dispersion relationship discussed in Section <ref>.

§ DISCUSSION

Observational studies of the stars and regions in IC 10 suggest that the starburst occurred relatively recently, likely in the past $\lesssim10$Myr <cit.>. Clustering of the young stellar population [Vacca et al., 2007] and similar properties across multiple regions of the starburst [Polles et al., 2019] indicate a potential common origin for the recent star formation.

In our KCWI observations of the regions in IC 10's central starbursting region, we find further evidence that these regions are young, with average crossing times $\rm<Myr$. Feedback plays an important role in the regulation of star forming environments and the dissipation of material. There are many forms this feedback can take and their contribution can be estimated from their energy or momentum input into the ISM. In our KCWI observations, we only observe the ionized gas and therefore cannot estimate the contribution of every form of potential feedback, but we do estimate the contribution of two important factors: direct radiation pressure, $\rm P_{dir}$, and warm gas pressure, $\rm P_{gas}$. Of these two we find the contribution of $\rm P_{gas}$ to be dominant over $\rm P_{dir}$ by $\sim$3 orders of magnitude. It should be noted that radiation pressure is sensitive to the effects of stochastic sampling of the initial mass function since the majority of the radiation is produced in the most massive stars. The impact of radiation pressure can therefore be somewhat uncertain in smaller clusters like those powering IC 10's regions where the stellar population of the ionizing cluster is not well represented by the assumed IMF. Though, since $\rm P_{gas}$ is $\sim$3 orders of magnitude greater than the estimated $\rm P_{dir}$ we do expect radiation pressure to be comparably negligible even if the estimate is impacted by stochasticity. The same trend with a minor contribution from $\rm P_{dir}$ is also found in the giant region 30 Doradus [Pellegrini et al., 2011, Lopez et al., 2011], a sample of 32 LMC and SMC regions [Lopez et al., 2014], and an additional sample 11 LMC regions [McLeod et al., 2019], although to a lesser extent.

The total $\rm P_{out}$ is on average $3\times$ the inward pressure which is predominantly due to turbulence in the surrounding gas. The self-gravity of the regions is comparably weak in the compact regions found in IC 10. This is in contrast to results found in the molecular ISM of nearby galaxies showing kpc and sub-kpc scale equilibrium between gravitational potential and outward pressure <cit.>. However, our regions observations are on much smaller size scales, resulting in the average $\rm P_{out} \sim 3\times P_{in}$,  and with 89% of the regions identified in IC 10 showing greater $\rm P_{out}$ indicating expansion. This is somewhat at odds with theoretical expectations of feedback effectiveness. For both direct radiation pressure and warm gas pressure, massive stars are expected to be the dominant source of the required ionizing photons, and thus these mechanisms are not expected to limit the star formation efficiency in populations with low stellar mass clusters <cit.>. This may vary by cluster due to stochasticity of the IMF, but given we find $\rm P_{gas}$ to be an effective counter to inward pressure in the majority of IC 10's regions this may require further exploration. We should note that discussions of limiting the star formation efficiency are usually explored in the context of the larger molecular cloud, while we are limited to studying these mechanisms within the regions and therefore cannot directly measure how much impact these mechanisms will have on clearing gas further from the ionizing cluster, but this still implies that $\rm P_{gas}$ may effectively limit star formation efficiency in a wider range of environments than previously expected. This is compounded by another potential source of expansion in the detection of diffuse X-ray emission by Wang et al., 2005 for which they note a morphological similarity with the   gas in the region of IC 10 that we observe with KCWI. They argue that this indicates the hot gas is still confined and may be driving expansion of the surrounding ionized gas structures. However, the effectiveness of this hot gas in driving feedback is dependent on leakage and mixing and likely sub-dominant compared to $\rm P_{gas}$ [Lopez et al., 2014].

The stacked spectrum of all the identified regions shows further evidence of expansion and/or outflowing gas, with an underlying broad component to the 5007  emission line with $\sigma\sim35$. Even the regions with evidence of rotation do not seem to have reached an equilibrium state with the surrounding gas as seen in the $\sim$14  higher measured velocity dispersions than what would be due to just rotation as well as virial parameters, $\rm \alpha_{vir} >>1$. The  higher velocity dispersions and elevated $\rm log \left([O{\sc III}]/H\beta\right)$ at the edges of many of the regions indicate that there may be shocks present which could be due to expansion of the region, champagne flows, or hot stellar winds mixing with cold gas. Over 35% of the regions show these pockets of elevated velocity dispersion at or near the edges of the region. The velocity dispersions are not high enough to be produced by shocks in the regions themselves as the elevated dispersions are $\sim$2$\times$ the sound speed, $c_s$. However, at the region boundaries the expanding gas may collide with cold molecular gas in the ISM which could induce a shock. With typical ISM temperatures of $\sim$100K, $c_{\rm s}$ would be on the order of 1  for an ideal gas; 20$\times$ less than the areas considered to have elevated velocity dispersion. A previous study of IC 10 ionized gas by Thurow & Wilcots, 2005 also found systematically larger line widths outside of regions, but they attribute this to superposition of different filaments or shells. These IFU observations have similar spectral resolution to our small slicer, R$\sim$18,000 observations, but with KCWI we are able to achieve 3$\times$ greater spatial resolution, limiting the potential for superposition of structures with different velocities. Given that we observe areas of higher velocity dispersion at spaxels with the highest $\rm log \left(\oiii5007\AA/H\beta\right )$ we find shocked gas to be a plausible alternative to the superposition of independent filaments, particularly with the significant improvement in spatial resolution of KCWI requiring these structures to be aligned along the line of sight on $\sim$few pc scales.

Resolved areas of significantly elevated velocity dispersion ($\sigma > 25$) were identified around 6 regions with average eddy turnover times $\rm \tau_{eddy}=l/\sigma \sim 0.2Myr$. The turbulent volumes around the three largest of these regions are most likely caused primarily by champagne flows, but stellar winds from early O stars could also provide sufficient energy to support the rate of turbulent dissipation ($\rm L_{turb}$). Two of the smaller regions with resolved turbulent volumes at the border are ionized by lower mass stars with therefore lower rates of ionizing photon production. In this case we find that stellar winds are more likely to support the observed turbulence and are again sufficient to support $\rm L_{turb}$. One of the six turbulent volumes has an estimated $\rm L_{turb}$ greater than what we estimate could be provided by the region and therefore is likely due to some external source of turbulence.

$\rm \tau_{eddy}$ measured for these turbulent regions is at the low end of the range observed in galactic scale outflows driven by starbursts <cit.>. However, the velocity measured in these ionized gas outflows, FWHM $\sim 61$, is significantly lower than that measured in galactic winds and outflows from individual star forming clumps in other studies. In a sample of 25 star-forming clumps located in LIRGs, Arribas et al., 2014 identify outflows in 83% of clumps with typical $\rm FWHM \sim 200$. Similarly, in the Rodríguez del Pino et al., 2019 study of ionized gas outflows in MaNGA galaxies, finding typical $\rm FWHM \sim 350$  in outflows originating from star-forming regions. Both of these studies find significantly higher velocities to the ionized gas outflows than the IC 10 regions which could be due to larger SFR (particularly in the case of the LIRGs), but they also have $\sim100\times$ lower spatial resolution to what is achieved in IC 10 with KCWI. These measured outflows may then be due to aggregate measurements of multiple compact regions and outflows. Some of this variation may also be due to biases introduced by differences in outflow detection methods. Both Arribas et al., 2014 and Rodríguez del Pino et al., 2019 rely on separating a broad outflow component in the emission line of the integrated star-forming region spectra while the resolution of this study allows direct detection of the influence of outflows in the ionized gas surrounding the regions. This may then simply probe more localized, lower velocity outflows than has been possible in previous studies.

The velocity dispersions measured inside the identified regions are also elevated relative to what would be expected for a rotating region in equilibrium for all cases showing evidence of rotational motion. The regions we identify in IC 10 tend to be offset from the scaling relationships found between the region size & luminosity as well as from the Larson's Law between size & velocity dispersion. Krieger et al., 2020 find a similar offset to larger line widths for a given size when comparing molecular clouds in the starburst NGC 253 to clouds in the Milky Way's Galactic center, which they attribute to gas that is not gravitationally bound and rather lies in transient structures. Based on the discrepancies in mass estimates and the young ages of IC 10's regions, it seems likely that these regions are still young enough to be undergoing expansion. They may then evolve onto the typical size-luminosity and size-velocity dispersion sequences after reaching an equilibrium state with the surrounding ISM.

The evolutionary stage of the Wolf-Rayet stars in the field of view provide another clue that the regions which host them are young. Wolf-Rayet stars are a later evolutionary stage of O stars which occurs before a type I SNe. In general, high mass O stars ($\rm >40M_{\odot}$) evolve to a WN type WR, then a WC, and then a SNIc, whereas lower mass O stars are thought to not reach the WC stage and simply explode as a SNIb  [Crowther, 2007]. Of the eight WR stars in the field covered by our KCWI observations, four of these fall in an region. Two have been identified as WN spectral types (M24 and T5 in regions H16a and M12 respectively), one is identified as an early WC (R10 in region I18), and the last has not yet been spectroscopically confirmed. Of the four WR stars outside of IC 10 regions, one is a late WN type and the rest are identified as WC. These stars are located in areas where ionized gas is nearby, but more diffuse and filamentary in appearance. The locations of these WR stars and their spectral type are shown in Figure <ref> with the observed 5007  flux and region contours. Though this is a small sample, the trend implies that WR stars in the WN stage could be more likely to be found in current regions, whereas WC types might be more likely found in areas of diffuse gas where an region may have previously been present. This would imply that the star clusters the WN stars belong to are younger and the surrounding gas has not yet been disrupted.

5007  integrated flux map with region contours reproduced from Figure <ref> with marked locations of all known WR stars ('X') and WR candidates ('+'). The stars designation is included to the right of the location marker with the WR type in parentheses if it has been spectroscopically confirmed.

§ SUMMARY

We made use of the highest resolution mode of the Keck Cosmic Web Imager IFS $-$ 0.35$\arcsec$ spatial sampling with 1$\arcsec$ FWHM and R$\sim$18,000 $-$ to study the population of regions in our nearest starburst galaxy IC 10. These high quality IFS observations allowed us to study the spatial and kinematic properties of the regions in detail. We identified 46 individual regions in the central burst of the irregular galaxy with a total SFR$\sim6\times10^{-3}$ . The average region identified has a size of 4.0 pc, an SFR of $\rm 1.3\times10^{-4}$ , an ionized gas mass $\rm M_{HII}\sim 56 \, M_{\odot}$, and a velocity dispersion of $\sim$16.

Over 95% of the identified region luminosities are consistent with the ionizing photon production rate of a single O or B star. 10 of the regions ($\sim$22%) can be powered by a single B star, with the lowest luminosity region requiring at minimum a star of spectral type B0.5.

The regions appear to be blue-shifted relative to the systemic velocity of IC 10 ($\sim$12) while the diffuse surrounding gas often shows a relative red-shift. Supplementary KCWI observations with lower resolution and wider wavelength coverage are used to estimate the oxygen abundance via the KK04 and PT05 metallicity calibrations, both of which make use of the $R_{23}$ strong-line calibration. These estimates yield averages of $\rm 12+log(O/H)_{PT05,lower}\approx7.85\pm0.03$ and $\rm 12+log(O/H)_{KK04,lower}\approx8.22\pm0.02$ with the “true" metallicity expected to lie within this range.

IC 10's regions have very short crossing times ($\rm \tau_{cr} < Myr$) and are not virialized ($\rm \alpha_{vir}>>1$ and $\rm M_{vir}>>M_{HII}$). The measured velocity dispersions in the regions are also too high to be due to rotational motion alone (by $\sim$11-12 ). We see evidence that these regions are generally still undergoing expansion. The IC 10 regions are offset from the scaling relationships found between the region size & luminosity as well as the size & velocity dispersion relationship. As these regions expand they may evolve onto the scaling relationships determined for the other samples of regions and star-forming clumps.

We estimated the contribution of the thermal gas pressure, $P_{gas}$, and direct radiation pressure, $\rm P_{dir}$, to the outward pressure in the regions. We find $\rm P_{gas}$ to be the dominant force of expansion in IC 10's regions, being $\sim$3 orders of magnitude greater than $\rm P_{dir}$ using the definition of $\rm P_{dir}$ based on ionizing photon production rate. We also find $\rm P_{out} > P_{in}$ in 89% of the regions before accounting for additional expansion from hot gas pressure, a somewhat surprising result given the low stellar masses estimated for the ionizing stars. Five of the regions show evidence of outflows that may be supported by energy in the ionizing cluster either in the form of stellar winds or champagne flows. These pressure and energy estimates add further evidence that the regions in IC 10 are young and undergoing expansion into the ISM and suggest that thermal gas pressure may be a more effective form of feedback than previously expected from low mass clusters.

These high resolution and SNR observations were possible in just 1.5 nights of Keck observing time. From just this short time we were able to obtain detailed kinematic and flux maps of a significant number of regions. With additional observations the remainder of IC 10's regions can be observed in the same modes, with deeper observations of the diffuse gas and the supplementary low resolution mode. More expansive coverage of regions and the DIG in IC 10 will allow for detailed study of the kinematic and ionization state differences between these unique regions of ionized gas. KCWI observations at high spectral resolving power of the remainder of IC 10 would also double the number of regions in this unique starburst environment in which the impact of different modes of feedback and outflows can be investigated.

A relatively small investment of time with optical IFS's such as KCWI and VLT/MUSE can quickly yield a large sample of local star-forming regions in a wide range of environments. The large field of view and moderate spectral resolving power (R$\sim$2000-4000) of MUSE provides an efficient tool for mapping ionization states of regions and the ISM, while the R$\sim$18,000 mode of KCWI can be leveraged for a detailed look at the gas kinematics in compact regions. Utilizing these powerful IFU's across a wide sample of star forming galaxies will allow a detailed and statistically significant study of how environmental conditions impact the effectiveness of feedback mechanisms and vice versa; and whether there are age and environmental dependencies when looking at the scaling relationships. Targeted IFU studies of this kind are already well underway and as a larger collective sample is built we will be able to better compare these results with theoretical predictions of feedback and inform new models.

The reduced data cubes used in this study are available by request to facilitate further study beyond the scope of this project.

We thank the anonymous referee for their comments which helped improve the quality of this paper. The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain.

astrodendro [Thomas et al., 2013],
Astropy [Astropy Collaboration et al., 2013, Price-Whelan et al., 2018],
IPython [Perez & Granger, 2007]
Matplotlib [Hunter, 2007],
NumPy [Harris et al., 2020],
pandas [The Pandas Development Team, 2016, Wes McKinney, 2010],
photutils [Bradley et al., 2019],
PyStan [Stan Development Team, 2017],
reproject [Robitaille et al., 2020]
SHAPE [Steffen et al., 2011]

§ REGION NAMING CONVENTION

The previous naming of IC 10 regions was developed by Hodge & Lee, 1990 where regions are assigned a number in order of increasing RA. If that region breaks up into smaller knots then a letter is added after the complex number (e.g., 111a). Throughout the literature this region identifier is typically preceded by either “HL90" or simply “HL" to indicate the origin of the identifier. This is a simple and clear way of tabulating the regions found in this early study but there are some significant shortcomings of this system now. One difficulty is in quickly identifying regions in the   maps as regions are only numbered based on their RA while there can be a large spread in Dec from one region to the next in the sequence. The other more problematic issue is that with better resolution and sensitivity one would expect to identify new regions and complexes breaking into more knots. When this occurs there is not a clear way in which to assign an identifier to these new regions. If the next number in the sequence is assigned to each new region there would no longer be a clear ordering based on RA, and reassigning numbers to each region with every new identification would make comparison between studies exceedingly difficult.

We have therefore proposed a new naming convention that we believe addresses these issues for our study and allows extension to future studies with even wider fields using the grid described in Section <ref>. This numbering scheme simplifies identification of nearby regions in both RA and Dec as well as extension to a larger FoV. This could be applied to the larger IC 10 envelope by increasing the numbering range in Dec. and extending the RA designation to double and/or negative lettering (e.g., AA or -A). Fainter regions may be identified in already occupied grid squares, but the next trailing letter in the sequence can be added as these should have lower luminosity than what is identified here.

Coordinate grid for proposed region naming scheme overlaid on an HST/ACS image showing the optical extent of the galaxy. regions are named based on the lettered column and numbered row corresponding to their center. This could be extended to the larger envelope of IC 10 by going to larger positive and negative numbers as well as double and/or negative lettering (e.g., AA or -A). The gold rectangle outlines the region of IC 10 which our KCWI observations fall in.

§ IMPACT OF RADIUS DEFINITION

The choice of how to define the radius of an region varies significantly between studies, and particularly between local and high-redshift studies.

In observational studies of local regions there is wide variation in the methods used to define the size of the region, with this often not being a critical aim of the study. For example, the Green Bank Telescope Region Discovery Survey (HRDS) measures the sizes of 441 regions in the Milky Way by taking the mean of the FWHM of Gaussians fit to the RA and Dec components of continuum observations [Anderson et al., 2011]. In the extensive CALIFA survey of over 26,000 extragalactic regions, a custom procedure called  [Sánchez et al., 2012] <cit.> is used to identify regions. This procedure is similar in methodology to  except that the maximum expected extent of regions is provided as an input constraint. The sizes of identified regions then tend towards a relatively uniform distribution [Sánchez et al., 2012], and the authors note that extracting reliable sizes was not a main goal of their methodology partly due to the resolution of their observations.

Studies of local GMCs typically use the second moments of the cloud structure to determine its properties [Heyer & Dame, 2015]. Often the geometric mean of the second moments of the cloud structure (in the direction of greatest elongation and perpendicular to that) is used to describe the RMS extent of a cloud, $\sigma_{\rm r}$. An empirical factor is then used to determine the radius of a spherical cloud, $R=\eta\sigma_{\rm r}$. This factor was first determined empirically in Solomon et al., 1987 to be $\eta=1.91$ for converting their rectangular regions to spherical clouds and is used throughout the literature to convert the second moments of a variety of structures to a spherical radius. Rosolowsky & Leroy, 2006 go through the derivation of this factor of $\eta$ for a spherical cloud with a density profile of $\rho\propto r^{-1}$ and determine a theoretical value of $\sqrt{6}\approx2.45$. They suggest that deviation may be due partly to the use of CO data to trace the GMC density which is shallower than the actual density profile due to saturation in dense regions and lack of detection in low density regions. This would mean that the “true" value of $\eta$ would lie somewhere between the empirical value of 1.91 from Solomon et al., 1987 and the value of 2.45 determined from their toy model of a GMC. However, they recommend continued use of the radius definition from Solomon et al., 1987 in order to remain consistent with this data set. Since our observations are of ionized rather than molecular gas we do not use this same factor of $\eta$, and instead assume a Gaussian profile in our determination of $r_{1/2}^*$ from the second moments.

Zaragoza-Cardiel et al., 2017 combine CO and   observations in local LIRGs with regions identified using . They define the radius, $r$, using the second moments of the structures in both cases, with the factor of 1.91 from Solomon et al., 1987. They compare the ratio of this radius to that derived from the area of the full structure, $r_{\rm eff}$, and find an average $\frac{r}{r_{\rm eff}}=0.86\pm0.14$. In another sample of local LIRGs with star-forming regions identified by , Larson et al., 2020 use $r_{\rm eff}$ to define the size of regions. Larson et al., 2020 also performs a comparison of the  and  identification routines finding similar average radii and SFR, but a narrower range of fluxes for a given radius of star-forming region due to a lack of local background subtraction.

In studies of high-redshift star-forming clumps, differences in the method used to determine region sizes are also present, although perhaps less significant due to the decrease in resolution. Wisnioski et al., 2012 compares the size determined from isophotes of constant flux and from fitting a Gaussian to the radial surface profile of regions. They find $r_{\rm eff}$ from the isophotal method to be systematically larger than $r_{1/2}$ determined via Gaussian fitting, but the luminosities to be consistent between the two methods. This comes from the emission being dominated by the higher intensity cores of the regions. They argue that $r_{1/2}$ from Gaussian fitting is a better defined observational parameter as it is less likely to be contaminated by diffuse emission that could be significant at high-redshift. Livermore et al., 2012 also compare clump sizes determined by $r_{\rm eff}$ of  isophotal structures and from fitting a 2D elliptical Gaussian profile to emission peaks. They find that using  for their sample gives 25% larger estimates of size than from the FWHM of the 2D elliptical Gaussian, but include error bars encompassing both measures. They note a smaller deviation between their  radii and the FWHM of fitted Gaussians than in Wisnioski et al., 2012 due in part to the use of multiple isophote levels in  that does not need to be tuned in the same way as other single isophote methods.

Which of the many possible radius definitions is used has a significant impact on the typical size of star-forming regions and on the scaling relationships determined from those properties. To quantify just how much impact the choice of using the pseudo half-light radius ($r_{1/2}^*$) definition for the region radii has in this study we perform MCMC fitting with different choices of radius definition and identification constraint for our sample. We fit the relationship between region size and luminosity for (i) the IC 10 regions alone, for (ii) local samples only, and for (iii) the full sample of local and high-redshift regions and clumps. These results are shown in Table <ref>.

lcccccccc [h]
Radius Definition Implications
Constraint     Radius Definition     Filtering     $\rm N_{regions}$     $\rm R_{avg}$     3cSlopes

(pc)     IC 10     Local     All Data
56em2r $>$ FWHM     $r_{1/2}^*$         46     4.0     $3.601^{+0.471}_{-0.385}$     $3.049^{+0.029}_{-0.025}$     $3.002^{+0.023}_{-0.024}$

$r_{1/2}^*$     r$>$FWHM      20     5.6     $2.972^{+0.793}_{-0.483}$     $3.147^{+0.032}_{-0.035}$     $3.067^{+0.032}_{-0.021}$

$r_{\rm eff}$         45     6.8     $2.979^{+0.426}_{-0.344}$     $3.045^{+0.025}_{-0.026}$     $2.987^{+0.020}_{-0.028}$

$r_{\rm maj}$         46     5.1     $3.176^{+0.451}_{-0.351}$     $2.999^{+0.036}_{-0.026}$     $2.951^{+0.031}_{-0.025}$

r $>$ FWHM     $r_{1/2}^*$         23     6.1     $2.888^{+0.558}_{-0.388}$     $3.161^{+0.026}_{-0.025}$     $3.043^{+0.025}_{-0.024}$

Results of model slope determined from MCMC fitting of the size-luminosity relationship for IC 10 regions and the local and high-redshift comparison sample based on different ways of defining and constraining the radius of IC 10 regions.

Col 1: constraint used in  to define an independent structure. The top section uses the more relaxed constraint that the diameter of the region must be larger than the FWHM determined from standard star observations (this always uses the definition of $r_{1/2}^*$ to match the Gaussian fit to the stars PSF). In the bottom row we require this diameter to be twice the standard star FWHM.

Col 2: radius definition used to determine region size; $r_{1/2}^*$, and $r_{\rm eff}$ used as described in Section <ref>, and $r_{\rm maj}$ is determined by converting the second moment in the direction of greatest elongation to the HWHM of a Gaussian.

Col 3: additional filtering applied to exclude regions from MCMC fits beyond the manual filtering of bad regions as described in Section <ref>

Col 4: Number of regions found in IC 10 based on these constraints and filtering.

Col 5: Average radius of IC 10 regions.

Col 6 - 8: slope and uncertainty determined from MCMC fitting of the size-luminosity relationship for IC 10 regions only (7), local regions only (8) and all local and high-redshift data (9).

Regardless of our chosen definition for region size, the slope of the local and full sample is consistent within $<5\%$. However, there is a larger difference in the slope of the IC 10 region sample alone. Using the most relaxed resolution constraint in  and $r_{1/2}^*$ produces a slope for which the nominal value deviates by $18\%$ from either a stricter resolution constraint (bottom row) or the definitions of $r_{\rm eff}$ and $r_{\rm maj}$ for the region size. Fitting only the IC 10 regions, however, leads to significantly larger uncertainties so the nominal slopes for each radius constraint are still consistent within the $1\sigma$ uncertainties.

§ SPECTRA THUMBNAILS

Spectral thumbnails of 5007  from all regions detected in the small slicer, R$\sim$18,000 observing mode. These lines are fit by a single Gaussian profile (except for J16a with a double Gaussian profile) shown in cyan with the residuals shown in grey below the associated spectrum. The spectra have been normalized to the peak of the 5007  line.

Figure <ref> cont'd.

§ REGION MAPS

Thumbnail maps of regions identified in our KCWI observations. (Left): Flux maps of the surrounding area. (Center Left): Velocity shift of spaxels within the region relative to the systemic velocity of the region. (Center Right): Velocity shift of the region and the surrounding gas. (Left): Velocity dispersion within the region and the surrounding gas. Regions of elevated velocity dispersion may be indicative of outflowing gas, particularly when correlated with a velocity shift relative to the surrounding gas.

Figure <ref> cont'd.

Figure <ref> cont'd.

Figure <ref> cont'd.

Figure <ref> cont'd.

Figure <ref> cont'd.

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