M101 Group | Ray Garner

The Peculiar "Bulge" of NGC 5474

Mon, 03 Mar 2025 00:00:00 +0000

The Mystery

Almost two years ago when I was writing up , I needed to write up a “Conclusions and Future Work” chapter. The conclusions were straightforward, but coming up with an idea of things I can do in the far off future was hard. I think any scientist would tell you that doing the science is usually the easy part; coming up with the idea in the first place is the hard part!

But throughout my thesis which focused primarily on the nearby giant spiral galaxy M101, known as the Pinwheel Galaxy, I kept turning my attention to its companion dwarf galaxies. The biggest of these doesn’t have a cute name, but is intead known by its catalog number, NGC 5474. And just like M101, NGC 5474 is peculiar! Optically, it looks like a typical dwarf spiral galaxy with loose spiral arms and numerous sites of star formation. That is, except for its central bulge being offset to the north!

This galaxy really caught my attention, but sadly, I didn’t have time for it in my thesis to do a deep dive on it as much as I had M101. This was a perfect candidate for some of that “future work,” to figure out what’s going on with that offset bulge. Recently, a performed some photometry of the bulge to look at resolve stellar populations, basically seeing what types of stars were in the bulge and the surrounding disk of NGC 5474. Based on that data, they proposed that the bulge of NGC 5474 could actually be a dwarf elliptical galaxy orbiting around the disk of NGC 5474 instead of being a bulge in the plane of the disk as is the case for normal galaxies.

How Can We Figure Out the Puzzle?

How can you tell that for sure? Well, you’d need spectroscopy. Spectroscopy is a tool that allows you to see how much light of each individual wavelength there is in an object called a spectrum. Astronomers (and physicists and chemists) have put a lot of work into understanding spectra and what sorts of objects produce the different wavelengths of light. For example, star-forming regions that are full of gas ionized by hot, young stars produce “emission lines”, things like ionized hydrogen, oxygen, nitrogen, sulfur, etc. Meanwhile, stars have elements in their stellar atmospheres that absorb light before it can leave the star, so stars have “absorption lines”, like hydrogen, magnesium, iron, calcium, etc.

So for our case of NGC 5474, if we can take the spectrum of the bulge area, we’ll see a spectrum likely composed of both emission lines and absorption lines. Thankfully, we’re not the first people to do this in general, so we know that the disks of galaxies are where the star-forming regions lie that produce the emission lines. Meanwhile, elliptical galaxies which don’t tend to have any ionized gas should produce mostly the absorption lines produced by populations of stars.

One more thing to consider. If an object is moving away from us, its light will be at wavelengths shifted from what we measure here on Earth in our labs. This is the well-known that you probably have lots of experience when an emergency vehicle passes you on the road. As it approaches you, the sounds waves coming from its sirens are squished together and you hear a higher pitch. As it races away from you, the sound waves are spread apart and you hear a lower pitch. Light waves do this too and we call this being blueshifted and redshifted, respectively.

What does that mean for our spectroscopy of the bulge and disk of NGC 5474? Well, if there is a wavelength offset between the absorption lines in the bulge and the emission lines in the disk, then that indicates a velocity offset as well. A velocity offset would be a strong indication that the bulge and disk are two separate objects!

All of this was proposed by that group of Italian astronomers in . Sadly, I did not have any spectroscopy of NGC 5474. There was one collaboration called ¹ that would have covered NGC 5474 in their spectroscopic survey, but they hadn’t released the data for this galaxy yet. Oh well. I’d have to go out and collect this spectroscopy at some far off date. Who knows when that’ll be? I wrote it up this idea and got to work defending my thesis.

The PINGS Data

Barely a year after my thesis, and after I had picked up everything to start my new postdoc position at Texas A&M, I figured I could try and answer this question. So I worked up the courage to cold email the head of the PINGS collaboration and see if they were planning on doing anything with their spectroscopic data of NGC 5474. Thankfully, the head, Fabián Rosales-Ortega was more than happy to share the data with me! Yes! I could now try and figure out what the nature of this peculiar offset bulge is.

What did this data look like? Well it’s a special kind of spectroscopic dataset called “integral field spectroscopy” or IFS. Basically, instead of throwing down a thin slit on a galaxy and being able to only collect the light in that slit, IFS is like taking a larger picture of a galaxy at each wavelength of light. The result is a “data cube”, a 2D picture of the galaxy with a spectrum in each pixel. That means that not only do we have spectral information, we could also see how the spectra vary across the galaxy!

Or at least, that’s the idea in theory. In practice, things can go wrong. The PINGS survey collected IFS data on many galaxies, but out of all of them NGC 5474 was the only one who’s data collection was compromised. This galaxy’s data was collected not by Fabián, but by the folks who manage the observatory. Due to no fault of their own, this meant that the accuracy of the data was not up to par across the entire galaxy. But! The region around the bulge was of good quality and that’s all we needed for this project.

Circular Apertures

Now, how do we actually extract the spectra and measure their (potentially offset) wavelengths? There were two basic methods that we tried. The first is to just sum up all of the light in some circular apertures centered on the bulge. This has the benefit of increasing the signal-to-noise ratio of the data. The second method is to use the entire 2D dataset to produce velocity maps, so we can more easily see where velocities are different and where they’re the same.

I’m not going to go into all of the details of how we actually do this (that’s what is for!), but I’ll show you some interesting results.

First up are the circular apertures. We did this in a series of circular apertures ranging from relatively small that encompassed only a fraction of the bulge up to a large aperture that gets the entire bulge. We did this because the smaller aperture minimizes “contamination” from the underlying disk while the largest aperture maximizes the signal-to-noise at the expense of containing more of the underlying disk. Below, you can see the spectra for each aperture in varying shades of blue and green.

The spectra of circular apertures centered on the bulge of NGC 5474. The top panel shows the spectra for the four apertures used with their radii indicated in the legend. The middle panel shows an example of the absorption spectrum overlayed in black for one of the apertures. The bottom panel shows the emission line spectrum produced after subtracting the absorption line spectrum.

In each spectra, you can see both absorption and emission lines. If we focus on the largest aperture which is shown by itself in the middle panel, you can more clearly see these two features. The black line is a fit to the stellar population and its absorption lines, those parts that have lower fluxes at a particular wavelength. You can also see the emission lines sitting on top of the absorption line spectrum. Subtracting away the absorption line spectrum gives you just the emission lines in the bottom panel. These lines are very familiar to astronomers who study star-forming regions. We see lines of ionized hydrogen, oxygen, nitrogen, sulfur, and even ionized neon!

Now we just need to estimate the velocity offset for the emission and absorption lines separately. Remember, the absorption lines are likely being produced by stars in the bulge, while the emission lines are being produced by star-forming regions in the disk. When we estimate the velocity of the absorption lines, we get approximately 230 km/s. Doing the same for the emission lines gives a velocity of approximately 255 km/s. That’s a velocity difference of 25 km/s!

Spatially Resolved Velocities

Alright, but that’s just for some circular apertures. If we use each pixel in the data, each containing its own spectrum, what do we get for the velocities across the entire datacube? I won’t bore you with the details, but we can produce velocity maps that show how the velocities in the absorption lines and emission lines change across the entire dataset.

From left to right: the stellar velocities, gas velocities, and velocity differences across NGC 5474. The top row of panels shows the velocities, while the bottom row shows their associated uncertainties. The red circle outlines the position of the bulge in each map.

In the above figure, we’ll focus on the top row (the bottom row contains the uncertainties on the velocity measurements). In all of the plots, the red circle is the position and size of the bulge. The left plot is the velocity of the stars, i.e., measured from the absorption lines. You can see that the bulge is pretty coherent, all around the same velocity of about 230 km/s. The middle plot shows the velocity of the ionized gas, i.e., measured from the emission lines. There’s a much larger area that is of the same velocity, but importantly this area contains the bulge at a velocity of about 250 km/s. Finally, the right plot shows the velocity difference and the bulge stands out clearly! Spatially resolved, the bulge has a velocity difference between 20 and 40 km/s, very similar to the result from the circular apertures!

Is This The Only Answer?

Now, just because we’ve measured a velocity difference of about 25 km/s, doesn’t mean these two objects are separate. We need to put this measurement in the context of similar measurements and see if any other explanation is plausible.

First, the gas and stars in normal galaxies don’t have a large velocity difference. For example, in of 30 isolated spiral galaxies, a velocity difference of only 2 km/s was found, ten times lower than what we’ve measured in NGC 5474.

The Cigar Galaxy, M82. Credits: NASA/ESA/Hubble Heritage Team/J. Gallagher/M. Mountain/P. Puxley

Maybe we’re measuring some sort of gaseous outflow from the disk? If you look at galaxies that have intense rates of star formation, like the Cigar Galaxy (M82), you see that gas is being pushed out perpendicular to the plane of the disk by rapid star formation and all those strong stellar winds. Since NGC 5474 is face-on and not edge-on like M82, maybe we’re just looking straight down the barrel of this funnel of gas. Well, NGC 5474 is not a starbursting galaxy. It has a meager star formation rate, and it doesn’t have an active supermassive black hole at the center that could push gas out either.

It does look like this velocity difference between the bulge (absorption lines) and disk (emission lines) is real and isn’t caused by some of the more common explanations. This galaxy, which has been seen as just one peculiar galaxy, looks like it is instead a superposition of two galaxies! The fact that we don’t see any debris from an interaction (no long streams of stars, no huge burst of star formation, no wildly distorted velocity maps) likely says that these two objects are in the process of weakly interacting with each other.

The larger question remains: how does this all connect to M101? I’ve buried the lede, but approximately 300 million years ago, . NGC 5474 swung by M101, close enough to distort M101 and create that “big wonky arm” as I have grown to call it. We thought that the interaction also might have shifted the bulge of NGC 5474 north, but our new results seem to indicate otherwise. Clearly that close passage with M101 didn’t unbind the bulge-disk pair of NGC 5474, or maybe the bulge is a more recent addition? Honestly, we don’t know for sure. All I know is that someone needs to do some dynamical modeling of this three-body interaction, something that is famously difficult.

That’s all I’ve got for you! I am very happy that I got to put some words down about this project and even happier that I got to investigate this peculiar little galaxy. This only goes to show that even the smallest galaxies are curiosities unto themselves!

PINGS stands for the PPak IFS Nearby Galaxies Survey, an acronym that contains two acronyms in it! ↩︎
Yep, I really did include this line as the opening quote to the final chapter of my thesis! ↩︎

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!