Thesis | Ray Garner

M101 Group: Galaxy Evolution in a Nearby Laboratory

Sun, 09 Aug 2026 00:00:00 +0000

I have focused on the M101 Group since graduate school. Because it is only a few megaparsecs away, this small, isolated galaxy group gives us a detailed look at how spiral galaxies form and change. We can study star formation in individual H II regions, map chemical abundances across a whole galactic disk, use stellar populations to trace spiral structure and interactions, and examine neighboring galaxies as part of the same dynamic system.

I started studying M101 by asking: how much star formation happens beyond the bright optical disks of galaxies? This question led me to explore how M101 has grown, changed chemically, responded to its surroundings, and interacted with its satellites, especially the unusual galaxy NGC 5474.

In this research, I have used deep, wide-field imaging, long-slit and integral field spectroscopy, nebular abundance measurements, stellar population modeling, and multiwavelength analysis. Each finding led to new questions. The M101 Group has been both a laboratory and a foundation for shaping how I study galaxy evolution.

Why M101?

M101, or the Pinwheel Galaxy, is a large, nearby spiral galaxy seen face-on. Its disk is noticeably asymmetric, featuring a “big wonky arm.” The outer spiral structure looks disturbed, and its largest satellite, NGC 5474, is also unusual, with a bright central bulge that is clearly offset from the center of its underlying disk.

These features have suggested for some time that M101 and NGC 5474 recently interacted. However, the system is puzzling: its shape looks disturbed, but other signs of a strong interaction are unexpectedly subtle.

These questions have motivated much of my research. What effects does a weak tidal interaction have on a large spiral galaxy? Where should we look for its signs: in current star formation, chemical abundances, stellar ages, spiral structure, or satellite dynamics? And how long do these signs last?

The M101 Group provides a nearby laboratory for studying spiral structure, satellite interactions, chemical evolution, and star formation. Image credit: Doublas Rollain - Astrobin

Searching for Star Formation Beyond the Optical Disk

For my first major M101 project, I used deep, wide-field narrowband imaging from the Burrell Schmidt telescope to search about 6 square degrees around the group for very faint areas of ongoing star formation.

We focused on Hα, Hβ, and [O III] emission lines and created a method to distinguish real emission-line sources apart from foreground stars and background galaxies. Our goal was to find isolated H II regions or new star-forming dwarf galaxies that could mark tidal material from an encounter between M101 and its companions.

The surprising result was largely a null detection. Nearly all of our candidates were associated with the extreme outer disk of M101 rather than a widespread intragroup population. Only two unusual objects were associated with other systems: an outlying H II region near NGC 5474 and a star-forming object belonging to a background galaxy NGC 5486.

A zoomed-in view of the detected regions near NGC 5474 (top row) and NGC 5486 (bottom row). Left: the continuum-subtracted Hα image. Right: the broadband B image, stretched to show the faint structure of the outer disks. The red circle represents the size of a galaxy mask. The orange circle is an object detected in all three narrowband images, while the green circle is an object detected in the Hα and [O III] image.Figure 9 of Garner et al. (2021)

Instead of being disappointing, this result changed my perspective on the group. If M101’s obvious asymmetries came from a recent encounter, it seems that event did not spread much star-forming gas into the surrounding area. Or, if it did, the Hα signals we looked for had already faded.

What This Work Demonstrated

Through this project, I gained experience with deep optical imaging, careful continuum subtraction, detecting low surface brightness sources, emission-line photometry, building catalogs, rejecting contaminants, following up with spectroscopy, and interpreting null results.

It also taught me an important lesson that has guided my research: not finding an expected signal can still tell us something physical. The lack of intragroup star formation helped narrow down possible interaction histories for the system and encouraged me to search for more subtle signs within the galaxies.

Chemical Evolution Across M101

If the interaction had not produced large numbers of isolated star-forming regions, perhaps its signature could instead be found in the chemical structure of M101’s disk.

Using an expanded Burrell Schmidt dataset including Hα, Hβ, [O III], and [O II], I measured strong-line oxygen abundances for 720 H II regions across M101, NGC 5474, and NGC 5477.

M101 shows the usual drop in metallicity as you move farther from the center, but the full two-dimensional data revealed a more complex situation. The gradient flattens in the outer disk, and H II regions there show clear azimuthal abundance patterns. These differences from a simple radial gradient mostly appear in parts of the disk that may have been shaped by M101’s interactions with NGC 5474.

The Kobulnicky & Kewley (2004) oxygen abundances of M101 fitted with a simple exponential (dashed black line) and a broken exponential (solid black line). Regions in the turn-around region (orange circles), upper branch (blue triangles), and lower branch (green inverted triangles) are distinguished. Outside 1.3R₂₃, we display both the upper- and lower-branch solutions connected by a dotted line for each individual H II region.Figure 10 of Garner et al. (2022)

What This Work Demonstrated

This project expanded my expertise in nebular astrophysics and chemical evolution, including reddening corrections, emission-line ratios, strong-line abundance diagnostics, metallicity gradients, uncertainty propagation, and spatial statistics.

More broadly, it taught me to connect local and global views of galaxies. Individual H II regions show local conditions, but hundreds of these measurements can reveal patterns in chemical enrichment and mixing across kiloparsec scales. Linking the physics of single nebulae to the evolution of their host galaxies became a key theme in my later work.

Spiral Dynamics Through Stellar Populations

The same narrowband dataset also held information I had not planned to use. Besides nebular emission, some filters captured age-sensitive stellar absorption features, like Balmer absorption and changes in the continuum near the Balmer and 4000 Å breaks. By comparing these measurements to stellar population models, I developed a way to estimate stellar ages across M101’s disk.

The resulting maps showed the expected trend: younger stars are found farther from the center, which fits with inside-out growth. More interestingly, the stellar ages change in a regular way across spiral arms. In the mostly undisturbed inner disk, these age patterns match what we expect from large-scale spiral shocks. Surprisingly, similar patterns also show up farther out, beyond the expected corotation radius of the main inner spiral.

I saw these results as evidence that M101’s seemingly organized structure may come from several dynamically linked spiral patterns, not just one rigidly rotating spiral density wave.

The equivalent width through the Hβ filters (W_Hβ) and [O II] filters (W₃₇₄₆) for all pixels in the main disk of M101 distinguished by environment: the inner disk in red, spiral arms in blue, and interarm regions in green.Figure 9 of Garner et al. (2024)

What This Work Demonstrated

For this project, I had to use an observational dataset for a new question it was not meant to answer. I developed techniques for modeling stellar populations, linked narrowband observations to synthetic spectra, made spatial age maps, and interpreted these results in terms of spiral dynamics.

This work expanded my research from studying present-day ionized gas to looking at the stellar fossil record. Gas emission shows what a galaxy is doing now, while stellar populations hold clues about its past. By combining both, we get a much clearer picture of galaxy evolution than by using either one alone.

NGC 5474: A Hidden Galaxy in Plain Sight

My dissertation led me to another question: if an encounter with NGC 5474 helped reshape M101, what happened to NGC 5474 itself?

NGC 5474 has a strange shape. Its bright central part, usually called its ‘bulge,’ is about a kiloparsec away from the center of its star-forming disk. Some have suggested that this part is not a bulge, but actually a separate dwarf galaxy seen in front of NGC 5474.

With PINGS PPak integral field spectroscopy, I measured the velocity of the stars in the supposed bulge and compared it to the ionized gas velocity in the surrounding disk. The two are offset by about 24 km/s, which gives kinematic evidence that they are separate systems.

An RGB color image of NGC 5474.Figure 1 of Garner et al. (2025)

This idea was supported by work I supervised with undergraduate researcher Liv Moody, who used the narrowband data to study stellar populations across NGC 5474. The offset bulge is made up mostly of old stars and is different from the surrounding disk. The galaxy’s outer stellar populations also show extra spatial structure that may relate to its interaction history.

This part of the project also marked a change in my role as a researcher. I moved from doing the analysis myself to designing research questions, developing methods, and mentoring another researcher through an independent project.

Returning to M101 with HETDEX

My current work brings me back to M101, but with a very different dataset. The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) observed M101 using the VIRUS integral field spectrograph, giving us complete spectroscopic coverage of M101’s disk for the first time.

I am using these spectra along with multiwavelength photometry to map the galaxy’s stellar ages, metallicities, star formation history, and both stellar and gas kinematics. This work brings my M101 research full circle. My first projects used individual H II regions to study the galaxy’s current state. Now, HETDEX lets me compare those signs to the history recorded in the overall stellar population.

The Broad Picture

Taken together, these projects show that galaxy interactions do not always leave just one clear observational sign.

In the M101 Group, signs of dynamical evolution look different depending on which tracer you study:

The surrounding group contains little widespread intragroup star formation;
The disk of M101 contains chemical structure beyond a simple radial gradient;
Its stellar populations encode both inside-out growth and complex spiral dynamics;
NGC 5474 contains a dynamically distinct central component and unusual stellar populations;
The stellar fossil record provides a way to connect these present-day signatures to longer evolutionary timescales.

This is one of the main lessons I have learned from these projects. Galaxy evolution is complex and happens on many scales and timescales. Features like shape, ionized gas, chemical abundances, stellar populations, and kinematics do not all react to an interaction in the same way or keep its signature for the same length of time. The M101 Group is especially valuable because its closeness lets us study all these tracers together.

Methods and Expertise

My work on the M101 Group has developed and demonstrated expertise across several areas of observational astronomy:

Observational Techniques: Deep wide-field imaging · Narrowband imaging · Long-slit spectroscopy · Integral field spectroscopy · Multiwavelength photometry

Galaxy Evolution: Star formation · Galaxy interactions · Spiral structure · Stellar populations · Chemical evolution

Nebular Astrophysics: H II region identification · Emission-line photometry · Reddening corrections · Strong-line abundances · Metallicity gradients

Stellar Populations: Synthetic stellar population modeling · Age-sensitive spectral features · Full spectral fitting · Star formation histories · Stellar metallicities

Data Analysis: Python · Astropy · CIGALE · pPXF · Spectral fitting · Image analysis · Catalog construction · Spatially resolved mapping · Statistical modeling

Selected Publications

First Author Work

Garner et al. 2026:
Spatially resolved stellar ages and metallicities from HETDEX spectroscopy.
Garner et al. 2025:
Kinematic evidence that the offset bulge of NGC 5474 is a distance stellar system.
Garner et al. 2024:
Using age-sensitive narrowband measurements to trace disk growth and spiral dynamics.
Garner et al. 2023:
Spectroscopic confirmation and interpretation of unusual objects discovered in the original narrowband survey.
Garner et al. 2022:
Spatially resolved chemical abundances across M101 and its major satellites.
Garner et al. 2021:
Deep search for star formation and low-luminosity galaxies throughout the M101 Group.

Collaborative & Mentored Work

Moody & Garner 2025:
Undergraduate-led project extending my stellar population methodology to NGC 5474. I mentored the project and contributed to the analysis and physical interpretation.
Watkins et al. 2024:
Collaborative investigation of the origin of diffuse ionized gas using the same deep imaging dataset together with ultraviolet observations and photoionization modeling.

Dissertation

My PhD dissertation brought together the originaly outlying star-formation, chemical abundances, and stellar population studies into a unified investigation of M101 and its group environment.

Where This Work Is Going

The M101 Group is still a key part of my research because it lets me connect physical processes measured on many different scales. The methods I developed here can be used beyond M101. Studying nearby galaxies allows us to link detailed nebular physics, stellar populations, chemical abundances, and galactic dynamics in ways we cannot with more distant galaxies. These local relationships can then help us understand galaxies across cosmic time.

Stellar Age Patterns Across the Disk of M101 with Narrowband Imaging

Tue, 05 Dec 2023 00:00:00 +0000

Investigating the Galactic Evolution of M101 with Deep Narrowband Imaging

Mon, 23 Oct 2023 14:30:00 +0000

PhD Thesis: Investigating the Galactic Evolution of M101 with Deep Narrowband Imaging

Fri, 14 Jul 2023 00:00:00 +0000

Investigating the Galactic Evolution of M101 with Deep Narrowband Imaging

Sat, 01 Jul 2023 00:00:00 +0000

Becoming Galactic Detectives!

Wed, 07 Jun 2023 00:00:00 +0000

Investigating the M101 Group with Deep Narrowband Imaging

Wed, 11 Jan 2023 00:00:00 +0000

Investigating the Oxygen Abundance of the M101 Group with Deep Narrowband Imaging

Wed, 09 Nov 2022 16:35:00 +0000

The M101 Interaction & Narrowband Photometry

Wed, 22 Jun 2022 10:00:00 +0000

Calculating Uncertainties on Strong-line Abundances

Tue, 02 Nov 2021 00:00:00 +0000

Background - Strong Line Abundances

When determining the oxygen abundance of an H II region or a star-forming galaxy as a whole, the most direct and physically motivated method is to measure the electron temperature ($T_e$) of the ionized gas using the intensity of one or more temperature-sensitive auroral lines such as [O III]λ4363, [N II]λ5755, [S III]λ6312, and [O II]λ7325 ( ; ; ). Unfortunately, these auroral lines are intrinsically faint, especially in metal-rich galaxies and star-forming regions. Therefore, when these lines are undetectable or not measured, strong-line abundance calibrations are used to estimate the metallicity of the ionized gas.

Strong-line abundance calibrations essentially relate the oxygen abundance to one or more line ratios involving the strongest recombination and collisionally excited (forbidden) lines (e.g., [O II]λ3727, Hα, Hβ, [O III]λλ4959,5007, [N II]λλ6548,6584, and [S II]λλ6716,6731). Strong-line methods are indirect and usually model dependent.

Over the last four decades, numerous strong-line calibrations have been developed, but in general they fall into one of three categories: semi-empirical, empirical, and theoretical. The older, semi-empirical calibrations were generally tied to electron temperature measurements at low metallicites and photoionization models at high metallicities (e.g., ; ; ). These hybrid calibrations were born from the observational difficulty of measuring the electron temperature of metal-rich H II regions. In contrast, the empirical methods are calibrated against high-quality observations of individual H II regions with measured direct (i.e., $T_e$-based) oxygen abundances (e.g., , ; ; ). One of the limitations of the empirical calibrations, especially in the metal-rich regime, is that they are based on observations of relatively small samples of high-excitation H II regions, whereas most integrated spectra of galaxies exhibit softer ionizing radiation fields. Finally, the class of theoretical abundance calibrations is based on ab initio photoionization model calculations, in which various nebular emission line ratios are tracked as a function of the input metallicity and ionization parameter (e.g., ; ).

The individual problems with these various calibrations is compounded by the fact that there exist large, poorly understood systematic discrepancies, in the sense that empirical calibrations generally yield oxygen abundances that are factors of 1.5-5 lower than abundances derived using theoretical calibrations ( ). Unfortunately, the physical origin of this systematic discrepancy remains unsolved.

Some of the more common strong-line calibrations rely on the metallicity-sensitive $R_{23}$ parameter ( ):

$$R_{23} = \frac{\text{[OII]}\lambda3727 + \text{[OIII]}\lambda\lambda4959,5007}{\text{H}\beta}.$$

The advantage of $R_{23}$ as an oxygen abundance diagnostic is that it is directly proportional to both principal ionization states of oxygen, unlike other diagnostics that have a second-order dependence on the abundance of other elements like nitrogen or sulfur. The disadvantages of $R_{23}$ are that it is sensitive to AGN contamination, and it must be corrected for stellar absorption and dust attenuation.

An additional complication is that the relation between $R_{23}$ and metallicity is famously double-valued. For example, using the calibration of , we can see the issue.

import matplotlib.pyplot as plt
import numpy as np
from astropy.table import Table
from astropy.io import fits
from scipy.stats import norm
%matplotlib inline

plt.rcParams.update({'font.size': 14, 'xtick.minor.visible':True,
 'ytick.minor.visible':True})
np.random.seed(5749)

def mcgaugh91_up(logR23,logO32):
 return 12 - 2.939 - 0.2*logR23 - 0.237*logR23**2 - 0.305*logR23**3 - 0.0283*logR23**4 - logO32*(0.0047 - 0.0221*logR23 - 0.102*logR23**2 - 0.0817*logR23**3 - 0.00717*logR23**4)

def mcgaugh91_dw(logR23,logO32):
 return 12 - 4.944 + 0.767*logR23 + 0.602*logR23**2 - logO32*(0.29 + 0.332*logR23 - 0.331*logR23**2)

fig1 = plt.figure(figsize=(8,6), constrained_layout=True)
ax1 = plt.subplot(111)

O32_vals = [-.5, 0, 0.5]
R23_vals = np.arange(-1.0,1.5,0.01)
z_vals = np.arange(6.5,9.5,0.01)
for j,ls in zip(O32_vals,['-','--','-.']):
 idx = np.argwhere(np.diff(np.sign(mcgaugh91_up(R23_vals,j) - mcgaugh91_dw(R23_vals,j)))).flatten()
 ax1.plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_up(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
 ax1.plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_dw(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
ax1.set_xlabel(r'$\log(R_{23})$')
ax1.set_ylabel(r'$12 + \log(\mathrm{O}/\mathrm{H})$')
ax1.set_xlim(-1.0,1.5)

In the above figure, we have the $R_{23}$ parameter on the $x$-axis and the oxygen abundance on the $y$-axis. The M91 calibration accounts for the ionization parameter in the form of $\log(\text{O}_{32})$, where $\text{O}_{32}$ is

$$\text{O}_{32} = \frac{\text{[OIII]}\lambda\lambda4959,5007}{\text{[OII]}\lambda3727},$$

and this characterizes the hardness of the ionizing radiation field; the styled lines indicate different blanket values of this O₃₂ parameter. However, we can see the broader issue. Metal-rich objects lie on the upper $R$₂₃ branch, while metal-poor galaxies and H II regions lie on the lower branch; the transition between the upper and lower $R$₂₃ branches is called the turn-around region. This non-monotonic relation between $R_{23}$ and $12+\log(\text{O}/\text{H})$ arises because in metal-rich star-forming regions, $R_{23}$ decreases with increasing metallicity as the far-infrared fine-structure lines (predominantly [O III]λ52μm and [O III]λ88μm) increasingly dominate the nebular cooling. At low metallicity, on the other hand, the optical transitions dominate the nebular cooling, and so $R_{23}$ decreases in tandem with decreasing metallicity because its strength is directly proportional to the abundance of oxygen atoms.

Because the $R$₂₃-metallicity relation is double-valued, we must take special care when computing the oxygen abundances of objects near the turn-around region, where the upper and lower branches intercept. In particular, objects that are statistically consistent with being on either the upper branch or the lower branch must have correspondingly large abundance errors. In addition, measurement uncertainties or residual AGN contamination occasionally result in an $R$₂₃ parameter that is larger than $\sim$10, the approximate theoretical limit for photoionization by massive stars ( ). Although these objects formally lie “off” the $R$₂₃ calibration, rather than rejecting them outright, which is typically what is done in the literature, it is better to asses whether they are statistically consistent with being on the $R$₂₃ calibration under consideration.

Here, I am going to illustrate and provide example code for utilizing the method outlined in to determine the oxygen abundance of star-forming regions in the spiral galaxy M101 and their associated uncertainties.

Background - Monte Carlo

I will refer any interested reader to the detailed example provided by Will Clarkson at the University of Michigan, which you can read by clicking . However, I will summarize a few brief points.

One of the most important pieces of model-fitting is to determine the “uncertainty” in the value of some parameter in the model. You might have fit some value of your model parameter to data, and it may even go through most of the datapoints and be consistent with your prior expectation on the parameter values. But unless you know what range of values of this parameter are consistent with the data, you really don’t know if your model fits at all.

First, a note and some assumptions.

It is common in the literature to condense the information contained in the deviations from the “best-fit” value to report the “1-sigma” range, often reported as $a \pm s$, where $s$ is the “1-sigma” range. In most cases, this means the range of values that bound 68 percent of the measured values under a large number of experiments. Formally, this practice throws away most of the information the reader might want to know: even under Gaussian measurement errors the posterior distribution of the best-fit parameter can be highly asymmetric and non-Gaussian. Simply reporting one number throws away the true distribution and is not good practice. However, it’s the practice most people go with, so we’ll try and take the baby before we throw out the bathwater.

I will additionally make the same assumptions that Will Clarkson made, namely that

our model $f(x)$ really does describe the behavior we are measuring, and
any deviations between the perfect underlying pattern predicted by $f(x)$ and those we measure, $y(x)$, are due only to measurement error that we can parameterize.

If we were to conduct a large number of identical experiments, then the “true” parameters of our model $f(x)$ would not change, but those are inaccessible to us; the parameters that best describe the data would change a little between experiments because we don’t measure exactly the underlying behavior. The range of best-fit values returned across the set of experiments is then a reasonable estimate for the range in the parameter values allowed by the data.

Since we cannot do an infinite number of repeat experiments, we need another way to predict what range of parameter values would be returned if we could do them. One way is the formal error-estimate: if the measurement errors all follow the same distribution, and if they are “small enough,” then you can use standard error propagation to take the measurement error and propagate it through to get a formal prediction on the error of the parameter. But there’s no guarantee that this will work in all cases. What you need in the real world is a method that will empirically find the range of parameters that fit the model to some level of “confidence” without actually doing ten thousand re-runs of the experiment to determine this range.

This is what Monte Carlo does in this context: simulate a large number of fake datasets and find the best-fit parameters using exactly the same method that you’re using to fit your real data. The range of returned parameters under these fake experiments is then a reasonable approximation to the true underlying error in the best-fit parameters.

An Example with Real Data

I have at my disposal emission line data for M101 taken from narrowband images of the M101 Group targeting the emission lines of Hα, Hβ, [O III]λλ4959,5007, and [O II]λ3727. The images were collected using Case Western Reserve University’s 24/36-in Burrell Schmidt telescope in Kitt Peak, Arizona. See and for an overview of some of the narrowband images and for an overview of our broadband capabilities.

Here, the data has been fully cleaned: Galactic extinction and interstellar reddening have been corrected, and line emission has been corrected for stellar Balmer absorption and contamination from [N II]λλ6548,6584 and [S II]λλ6716,6731 emission.

First, we load in the data, calculate the $R_{23}$ and O$_{32}$ line ratios, and their associated errors.

hdul = fits.open('M101_H&nbsp;<span style='font-size: 0.8em;'>II</span>_regions_corrected_photometry.fits.gz')
M101Data = Table(hdul[1].data)

M101Data['R23'] = (M101Data['O2_nored'] + M101Data['O3_nored'])/M101Data['Hb_nored']
M101Data['R23_err'] = np.abs(M101Data['R23'])*np.sqrt((np.sqrt(M101Data['O3_nored_err']**2 + M101Data['O2_nored_err']**2)/(M101Data['O2_nored']+M101Data['O3_nored']))**2 + (M101Data['Hb_nored_err']/M101Data['Hb_nored'])**2)
M101Data['O32'] = M101Data['O3_nored']/M101Data['O2_nored']
M101Data['O32_err'] = np.abs(M101Data['O32'])*np.sqrt((M101Data['O3_nored_err']/M101Data['O3_nored'])**2 + (M101Data['O2_nored_err']/M101Data['O2_nored'])**2)

Let’s take a look at the distribution of the data:

fig2 = plt.figure(figsize=(10,6), constrained_layout=True)
ax2 = fig2.subplot_mosaic(
 """
 abc
 def
 """)

for line, ax in zip(['Ha','Hb','R23','O3','O2','O32'],['a','b','c','d','e','f']):
 if line is 'R23' or line is 'O32':
 ax2[ax].hist(np.log10(M101Data[line]),bins=50,ec='k')
 ax2[ax].set_xlabel(r'$\log({0})$'.format(line))
 ax2[ax].set_ylabel(r'$N$')
 else:
 ax2[ax].hist(np.log10(M101Data[line+'_nored']),bins=50,ec='k')
 ax2[ax].set_xlabel(r'$\log(\mathrm{{{0}}})$ [erg s$^{{-1}}$ cm$^{{-2}}$]'.format(line))
 ax2[ax].set_ylabel(r'$N$')

We see that our line fluxes are across the same range and are somewhat Gaussian. For context, the Hα flux tapers off at around $10^{-15}$ erg s$^{-1}$ cm$^{-2}$ which corresponds to a star formation rate of $3 \times 10^{-5}$ M$_\odot$ yr$^{-1}$ ( ; ).

We can also do the same for the associated uncertainties:

fig3 = plt.figure(figsize=(10,6), constrained_layout=True)
ax3 = fig3.subplot_mosaic(
 """
 abc
 def
 """)

for line, ax in zip(['Ha','Hb','R23','O3','O2','O32'],['a','b','c','d','e','f']):
 if line is 'R23' or line is 'O32':
 ax3[ax].hist(np.log10(M101Data[line+'_err']),bins=50,ec='k')
 ax3[ax].set_xlabel(r'$\log({0})$'.format(line))
 ax3[ax].set_ylabel(r'$N$')
 else:
 ax3[ax].hist(np.log10(M101Data[line+'_nored_err']),bins=50,ec='k')
 ax3[ax].set_xlabel(r'$\log(\mathrm{{{0}}})$ [erg s$^{{-1}}$ cm$^{{-2}}$]'.format(line))
 ax3[ax].set_ylabel(r'$N$')

We see that our O3 and O2 errors are roughly Gaussian, while our Hα and Hβ error distributions might be Lorentzian in nature. Notably, the $R_{23}$ and O$_{32}$ errors do appear to be broadly Gaussian in nature. This is all good!

Now we can plot our $R_{23}$ values against the $R_{23}$-metallicity relation. However, to do that, we need some way of picking which branch a particular H II region lies on. The current recommendations for assigning branches to H II regions is to utilize line ratios with [N II] ( ; ). In that case, a region is assigned to the lower branch if $\log(\text{[NII]}/\text{H}\alpha) < -1$ and $\log(\text{[NII]}/\text{[OII]}) < -1.05$, while regions are assigned to the upper branch if $\log(\text{[NII]}/\text{H}\alpha) > -1$ and $\log(\text{[NII]}/\text{[OII]}) > -0.8$.

However, due to the placement of our narrowband filters, we have no way of measuring the [N II] line emission. Therefore, at the suggestion of , we use the calibration of , which does not require an upper/lower branch distinction to be made, to calculate an initial guess of the metallicity. This step is problematic for many reasons:

the Z94 calibration is itself an average of three older theoretical calibrations and will suffer from the same uncertainties in those calibrations;
the Z94 calibration will be very inaccurate at low abundances ($\log(\text{O/H}) + 12 < 8.5$), causing metallicities to be overestimated by a factor of up to 0.5 dex in this range; and
the Z94 calibration will preferentially assign most of the $\log(R_{23})$ parameter space to the upper branch, causing spurious artifacts to remain, such as a plateau in the radial abundance gradient (see for a discussion).

These problems notwithstanding, we will go ahead and use the Z94 calibration to make our initial guesses as to which branch each H II region belongs.

def zaritsky94(log_R23):
 return 9.265 - 0.33*log_R23 - 0.202*log_R23**2 - 0.207*log_R23**3 - 0.333*log_R23**4

def mcgaugh91(logR23,logO32,Z_guess):
 if Z_guess <= 8.4:
 return 12 - 4.944 + 0.767*logR23 + 0.602*logR23**2 - logO32*(0.29 + 0.332*logR23 - 0.331*logR23**2)
 elif Z_guess > 8.4:
 return 12 - 2.939 - 0.2*logR23 - 0.237*logR23**2 - 0.305*logR23**3 - 0.0283*logR23**4 - logO32*(0.0047 - 0.0221*logR23 - 0.102*logR23**2 - 0.0817*logR23**3 - 0.00717*logR23**4)

Z_guess = zaritsky94(np.log10(M101Data['R23']))
Z_real = np.zeros(0)
for i in range(0,len(M101Data)):
 Z = mcgaugh91(np.log10(M101Data[i]['R23']),np.log10(M101Data[i]['O32']),Z_guess[i])
 Z_real = np.append(Z_real,Z)

fig4 = plt.figure(figsize=(8,6), constrained_layout=True)
ax4 = plt.subplot(111)

O32_vals = [-.5, 0, 0.5]
R23_vals = np.arange(-1.0,1.3,0.01)
z_vals = np.arange(6.5,9.5,0.01)
for j,ls in zip(O32_vals,['-','--','-.']):
 idx = np.argwhere(np.diff(np.sign(mcgaugh91_up(R23_vals,j) - mcgaugh91_dw(R23_vals,j)))).flatten()
 ax4.plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_up(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
 ax4.plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_dw(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
ax4.plot(np.log10(M101Data['R23']), Z_real, color='C0', marker='.', markersize=5, ls='None')
ax4.set_xlabel(r'$\log(R_{23})$')
ax4.set_ylabel(r'$12 + \log(\mathrm{O/H})_{\mathrm{M91}}$')
ax4.set_xlim(-1,1.5)
ax4.set_ylim(8.0,9.25)

Once again, we have the $R_{23}$ parameter on the $x$-axis and the metallicity on the $y$-axis. The styled lines are the M91 calibration for particular O$_{32}$ values and the blue points are the H II regions in M101. Most of the regions fall on the $R_{23}$ calibration, but there are many regions with $\log(R_{23}) > 1.0$ that fall off of the calibration. Since the calibration is just an equation, it will spit out a number regardless of how unphysical that number is; in this case, the M91 calibration becomes unphysical for $\log(R_{23}) > 1.0$.

Normally, we would toss these high $R_{23}$-valued regions, and most of the literature does just that. However, you can imagine that if we were to plot the $R_{23}$ error bars as well, some of those “unphysical” regions could conceivable lie on the calibration given the full range of their uncertainties. This is where the Monte Carlo method devised by comes in.

We need to set up a few things first: the number of trials and the combined set of best-fit parameters for all the model parameters (initially empty).

n = 500
m101_z_up = np.array([]) # an array for the upper branch metallicities
m101_z_dw = np.array([]) # an array for the lower (down) branch metallicities

Now we do the simulations. Each time we need to generate the data as well as pass it through the calibrations, one solution for the upper branch and one solution for the lower branch.

for i in range(n):
 Hb_trial = np.random.normal(M101Data['Hb_nored'],np.abs(M101Data['Hb_nored_err']))
 O2_trial = np.random.normal(M101Data['O2_nored'],np.abs(M101Data['O2_nored_err']))
 O3_trial = np.random.normal(M101Data['O3_nored'],np.abs(M101Data['O3_nored_err']))

 O32_trial = np.log10(O3_trial/O2_trial)
 R23_trial = np.log10((O3_trial + O2_trial)/Hb_trial)

 # We use a try/except clause to catch weird regions
 try:
 up = mcgaugh91_up(R23_trial,O32_trial)
 except:
 dumdum = 1
 continue

 try:
 dw = mcgaugh91_dw(R23_trial,O32_trial)
 except:
 dumdum = 1
 continue

 # Stack the trial onto the running sample
 if np.size(m101_z_up) < 1:
 m101_z_up = np.copy(up)
 else:
 m101_z_up = np.vstack((m101_z_up, up))

 if np.size(m101_z_dw) < 1:
 m101_z_dw = np.copy(dw)
 else:
 m101_z_dw = np.vstack((m101_z_dw, dw))

A few points to note:

The routine might not always work. A more sophisticated analysis would catch those errors. Here, I’m using Python’s “try/except” clause to ignore the bad cases.
In this example, I am starting with an empty array and then stacking on the fit-values only if the fitting routine ran without failing. The “continue” statement stops the routine from dumbly stacking on the last fit-value if the fit failed. I do things this way so that the fitpars array is always the correct size to match the number of correctly-run trials.
I am picking random values surrounding each region’s values within its uncertainties, assuming a Gaussian distribution within those uncertainties. That may not necessarily be the case, but it will work for our purposes.

Having done all that, let’s look at the size of the set of trials:

print(m101_z_up.shape)
print(m101_z_dw.shape)

(500, 849)
(500, 849)

This shows that all of our 500 trials were successful! Now let’s take a look at the distribution:

print('Upper Branch Metallicity: {:.2f} +/- {:.2f}'.format(np.nanmedian(m101_z_up),np.nanstd(m101_z_up)))
print('Lower Branch Metallicity: {:.2f} +/- {:.2f}'.format(np.nanmedian(m101_z_dw),np.nanstd(m101_z_dw)))

Upper Branch Metallicity: 8.54 +/- 1.35
Lower Branch Metallicity: 8.15 +/- 0.55

Now comes the method outlined by Moustakas et al. (2010). You can imagine that there are three possible cases when you look at a particular H II region’s upper and lower branch solutions. In the first case, the upper and lower branch solutions are distinct and well-separated, i.e., they are outside of each other’s uncertainties. Once the appropriate branch has been chosen, the corresponding metallicity follows. This is easily seen in the left-hand plot below: the upper branch solution (red circle) is above and well-separated from the lower branch solution (blue triangle). The right-hand plot below shows the Monte Carlo distribution of $12 + \log(\text{O/H})$ values corresponding to the solution on the upper (solid red histogram) and lower branch (dashed blue histogram) assuming Gaussian errors on the [O II], [O III], and Hβ line flux after 500 trials. Assuming this object belongs on the upper branch, the $1\sigma$ uncertainty on $12 + \log(\text{O/H})$ is given by the width of the solid red histogram.

i = 3

fig5 = plt.figure(figsize=(14,7), constrained_layout=True)
ax5 = fig5.subplot_mosaic(
 """
 ab
 """)

# Plot the M91 calibration for particular O32 values
O32_vals = [-.5, 0, 0.5]
R23_vals = np.arange(-0.3,1.3,0.01)
z_vals = np.arange(6.5,9.5,0.01)
for j,ls in zip(O32_vals,['-','--','-.']):
 idx = np.argwhere(np.diff(np.sign(mcgaugh91_up(R23_vals,j) - mcgaugh91_dw(R23_vals,j)))).flatten()
 ax5['a'].plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_up(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
 ax5['a'].plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_dw(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
# Plot the upper branch solution
ax5['a'].errorbar(np.log10(M101Data[i]['R23']), np.nanmedian(m101_z_up[:,i]),
 xerr = (0.434*M101Data[i]['R23_err'])/M101Data[i]['R23'],
 yerr = np.nanstd(m101_z_up[:,i]), mec='r', mfc='r', marker='o', markersize=10, ecolor='r', elinewidth=2,
 capsize = 5, mew=2)
# Plot the lower branch solution
ax5['a'].errorbar(np.log10(M101Data[i]['R23']), np.nanmedian(m101_z_dw[:,i]),
 xerr = (0.434*M101Data[i]['R23_err'])/M101Data[i]['R23'],
 yerr = np.nanstd(m101_z_dw[:,i]), mec='b', mfc='b', marker='^', markersize=10, ecolor='b', elinewidth=2,
 capsize = 5, mew=2)
# Plot the upper and lower branch histograms
ax5['b'].hist(m101_z_up[:,i], bins=25, histtype='step', color='r', ls='-', lw=2)
ax5['b'].hist(m101_z_dw[:,i], bins=25, histtype='step', color='b', ls='--', lw=2)
# Set plot kwargs
ax5['a'].set_xlabel(r'$\log(R_{23})$')
ax5['a'].set_ylabel(r'$12 + \log(\mathrm{O}/\mathrm{H})_{\mathrm{M91}}$')
ax5['b'].set_xlabel(r'$12 + \log(\mathrm{O}/\mathrm{H})_{\mathrm{M91}}$')
ax5['b'].set_ylabel(r'$N$')
ax5['a'].set_xlim(-0.3,1.3)
ax5['a'].set_ylim(6.5,9.5)
ax5['b'].set_xlim(6.5,9.5)

# If upper branch solution is above lower branch solution
if np.nanmedian(m101_z_up[:,i]) > np.nanmedian(m101_z_dw[:,i]):
 # If they aren't well-separated, i.e., within each other's 1*sigma
 if (np.nanmedian(m101_z_up[:,i]) - np.nanstd(m101_z_up[:,i]) < np.nanmedian(m101_z_dw[:,i])) and (np.nanmedian(m101_z_dw[:,i]) + np.nanstd(m101_z_dw[:,i]) > np.nanmedian(m101_z_up[:,i])):
 (mu1, sig1) = norm.fit(m101_z_up[:,i][~np.isnan(m101_z_up[:,i])])
 (mu2, sig2) = norm.fit(m101_z_dw[:,i][~np.isnan(m101_z_dw[:,i])])
 avg = np.nanmean([np.nanmedian(m101_z_up[:,i]), np.nanmedian(m101_z_dw[:,i])])
 sig = np.nanmedian([np.nanstd(m101_z_up[:,i]),np.nanstd(m101_z_dw[:,i])])
 ax5['b'].axvline(avg, color='k', ls='-', lw=2, label=r'$12 + \log(\mathrm{{O}}/\mathrm{{H}}) = {:.2f} \pm {:.2f}$'.format(avg,sig))
 ax5['b'].legend(loc='best')
 # If they are well-separated, let the initial guess decide which branch to use
 else:
 if Z_guess[i] <= 8.4:
 ax5['b'].axvline(np.nanmedian(m101_z_dw[:,i]), color='k', ls='-', lw=2, label=r'$12 + \log(\mathrm{{O}}/\mathrm{{H}}) = {:.2f} \pm {:.2f}$'.format(np.nanmedian(m101_z_dw[:,i]),np.nanstd(m101_z_dw[:,i])))
 ax5['b'].legend(loc='best')
 else:
 ax5['b'].axvline(np.nanmedian(m101_z_up[:,i]), color='k', ls='-', lw=2, label=r'$12 + \log(\mathrm{{O}}/\mathrm{{H}}) = {:.2f} \pm {:.2f}$'.format(np.nanmedian(m101_z_up[:,i]),np.nanstd(m101_z_up[:,i])))
 ax5['b'].legend(loc='best')
# If upper branch solution is below lower branch solution
elif np.nanmedian(m101_z_up[:,i]) < np.nanmedian(m101_z_dw[:,i]):
 # If they aren't well-separated, i.e., within each other's 1*sigma
 if (np.nanmedian(m101_z_up[:,i]) + np.nanstd(m101_z_up[:,i]) > np.nanmedian(m101_z_dw[:,i])) and (np.nanmedian(m101_z_dw[:,i]) - np.nanstd(m101_z_dw[:,i]) < np.nanmedian(m101_z_up[:,i])):
 (mu1, sig1) = norm.fit(m101_z_up[:,i][~np.isnan(m101_z_up[:,i])])
 (mu2, sig2) = norm.fit(m101_z_dw[:,i][~np.isnan(m101_z_dw[:,i])])
 avg = np.nanmean([np.nanmedian(m101_z_up[:,i]), np.nanmedian(m101_z_dw[:,i])])
 sig = np.nanmedian([np.nanstd(m101_z_up[:,i]),np.nanstd(m101_z_dw[:,i])])
 ax5['b'].axvline(avg, color='k', ls='-', lw=2, label=r'$12 + \log(\mathrm{{O}}/\mathrm{{H}}) = {:.2f} \pm {:.2f}$'.format(avg,sig))
 ax5['b'].legend(loc='best')
 # If they are well-separated, mark it as indeterminant
 else:
 ax5['b'].text(0.05,0.9,'No determination possible', ha='left', va='center', transform=ax5['b'].transAxes)

In the second case, the formal solution on the lower branch is larger than the solution on the upper branch (note that the blue triangle is now above the red circle in the plot on the left below). In the right plot below, the overlapping region of the histograms corresponds to values of $R_{23}$ and O$_{32}$ that lie on the M91 calibration, that is, where $(\text{O/H})_{\text{upper}} > (\text{O/H})_{\text{lower}}$. If the central values of the two distributions are within $1\sigma$ of each other, as measured by the width of the overlapping region, we adopt the average of the two solutions as the oxygen abundance, and the width of the overlapping region as the $1\sigma$ uncertainty. In practice, these types of regions all have M91 abundances equal to the abundance around the turn-around region, $\approx 8.4$ dex; however, they also have large abundance errors, which reflects this ambiguity in picking a branch.

Finally, the third case is where no oxygen abundance measurement is possible using the M91 calibration. Here, the upper and lower branch solutions are statistically inconsistent with each other, given the measurement uncertainties; therefore no solution exists, and these objects must be rejected.

Now, we can write a code that doesn’t just plot this, but saves the determined (or undetermined) value for each region as a new column in our DataFrame.

M91 = np.zeros(0)
M91_err = np.zeros(0)
Z_guess = zaritsky94(np.log10(M101Data['R23']))
for i in range(0,len(M101Data)):
 if np.nanmedian(m101_z_up[:,i]) > np.nanmedian(m101_z_dw[:,i]):
 if (np.nanmedian(m101_z_up[:,i]) - np.nanstd(m101_z_up[:,i]) < np.nanmedian(m101_z_dw[:,i])) and (np.nanmedian(m101_z_dw[:,i]) + np.nanstd(m101_z_dw[:,i]) > np.nanmedian(m101_z_up[:,i])):
 (mu1, sig1) = norm.fit(m101_z_up[:,i][~np.isnan(m101_z_up[:,i])])
 (mu2, sig2) = norm.fit(m101_z_dw[:,i][~np.isnan(m101_z_dw[:,i])])
 avg = np.nanmean([np.nanmedian(m101_z_up[:,i]), np.nanmedian(m101_z_dw[:,i])])
 sig = np.nanmedian([np.nanstd(m101_z_up[:,i]),np.nanstd(m101_z_dw[:,i])])
 M91 = np.append(M91,avg)
 M91_err = np.append(M91_err,sig)
 else:
 if Z_guess[i] <= 8.4:
 M91 = np.append(M91,np.nanmedian(m101_z_dw[:,i]))
 M91_err = np.append(M91_err,np.nanstd(m101_z_dw[:,i]))
 else:
 M91 = np.append(M91,np.nanmedian(m101_z_up[:,i]))
 M91_err = np.append(M91_err,np.nanstd(m101_z_up[:,i]))
 elif np.nanmedian(m101_z_up[:,i]) < np.nanmedian(m101_z_dw[:,i]):
 if (np.nanmedian(m101_z_up[:,i]) + np.nanstd(m101_z_up[:,i]) > np.nanmedian(m101_z_dw[:,i])) and (np.nanmedian(m101_z_dw[:,i]) - np.nanstd(m101_z_dw[:,i]) < np.nanmedian(m101_z_up[:,i])):
 (mu1, sig1) = norm.fit(m101_z_up[:,i][~np.isnan(m101_z_up[:,i])])
 (mu2, sig2) = norm.fit(m101_z_dw[:,i][~np.isnan(m101_z_dw[:,i])])
 avg = np.nanmean([np.nanmedian(m101_z_up[:,i]), np.nanmedian(m101_z_dw[:,i])])
 sig = np.nanmedian([np.nanstd(m101_z_up[:,i]),np.nanstd(m101_z_dw[:,i])])
 M91 = np.append(M91,avg)
 M91_err = np.append(M91_err,sig)
 else:
 M91 = np.append(M91,np.nan)
 M91_err = np.append(M91_err,np.nan)
 else:
 M91 = np.append(M91, np.nan)
 M91_err = np.append(M91_err, np.nan)

M101Data['M91'] = M91
M101Data['M91_err'] = M91_err

Let’s take a look at the $R_{23}$-metallicity relation once more to see how we’ve faired.

fig4 = plt.figure(figsize=(8,6), constrained_layout=True)
ax4 = plt.subplot(111)

O32_vals = [-.5, 0, 0.5]
R23_vals = np.arange(-1.0,1.3,0.01)
z_vals = np.arange(6.5,9.5,0.01)
for j,ls in zip(O32_vals,['-','--','-.']):
 idx = np.argwhere(np.diff(np.sign(mcgaugh91_up(R23_vals,j) - mcgaugh91_dw(R23_vals,j)))).flatten()
 ax4.plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_up(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
 ax4.plot(R23_vals[R23_vals <= R23_vals[idx]], mcgaugh91_dw(R23_vals[R23_vals <= R23_vals[idx]],j), 'k', ls=ls)
ax4.plot(np.log10(M101Data['R23']), M101Data['M91'], color='C0', marker='.', markersize=5, ls='None')
ax4.set_xlabel(r'$\log(R_{23})$')
ax4.set_ylabel(r'$12 + \log(\mathrm{O/H})_{\mathrm{M91}}$')
ax4.set_xlim(-1,1.5)
ax4.set_ylim(8.0,9.25)

Comparing this plot to the one we made before, we have now removed the “unphysical” regions that did not fall on the calibration even within their uncertainties. There are four outliers which will require further investigation to determine whether they are realistic or not. Another feature we notice now is a line of points at about $12 + \log(\text{O/H}) \sim 8.4$. These are points in the turn-around region that have been “pushed” to the turn-around region metallicity. While this appears bad and might be a downside to the calibration procedure, remember that those points will have relatively higher uncertainties associated with them. When plotting a radial gradient, for instance, any line fitter that can take into account uncertainties should not be weighed heavily by these points.

Conclusions

Using Monte Carlo and the methodology outlined by Moustakas et al. (2010), I have hopefully illustrated its usefulness in robustly determining metallicities. This method allows for a straightforward calculation of realistic uncertainties on the abundance as determined by strong-line methods. Obviously, this method will still suffer from the known problems associated with strong-line abundance determinations, but we will be able to state our uncertainties in a more statistical sense. Additionally, although not illustrated here, this can be applied to any strong-line calibration using the $R_{23}$ parameter, not just M91.

As always, let me know if you spot any errors in my code or the logic behind this method!

iPoster @ AAS

Sun, 10 Jan 2021 00:00:00 +0000

Thesis | Ray Garner

M101 Group: Galaxy Evolution in a Nearby Laboratory

Why M101?

Searching for Star Formation Beyond the Optical Disk

What This Work Demonstrated

Related Publications

Chemical Evolution Across M101

What This Work Demonstrated

Related Publications

Spiral Dynamics Through Stellar Populations

What This Work Demonstrated

Related Publications

NGC 5474: A Hidden Galaxy in Plain Sight

Related Publications

Returning to M101 with HETDEX

Related Publications

The Broad Picture

Methods and Expertise

Selected Publications

First Author Work

Collaborative & Mentored Work

Dissertation

Where This Work Is Going

Stellar Age Patterns Across the Disk of M101 with Narrowband Imaging

Investigating the Galactic Evolution of M101 with Deep Narrowband Imaging

PhD Thesis: Investigating the Galactic Evolution of M101 with Deep Narrowband Imaging

Investigating the Galactic Evolution of M101 with Deep Narrowband Imaging

Becoming Galactic Detectives!

Investigating the M101 Group with Deep Narrowband Imaging

Investigating the Oxygen Abundance of the M101 Group with Deep Narrowband Imaging

The M101 Interaction & Narrowband Photometry

Calculating Uncertainties on Strong-line Abundances

Background - Strong Line Abundances

Background - Monte Carlo

An Example with Real Data

Conclusions

iPoster @ AAS