Emission Lines | Ray Garner

The N/O-O/H relation across different large-scale environments

Thu, 08 Oct 2026 00:00:00 +0000

The Helix Nebula as seen by the Local Volume Mapper

Thu, 21 May 2026 00:00:00 +0000

One of the big surveys in astronomy is the . Although it’s “old” by today’s standards, having started data collection in the far-off year of 2000, this survey was groundbreaking. It would survey the sky and take pictures in several different filters, resulting in the imaging of billions of objects. They also followed up with spectroscopy, taking the light from a distant object and splitting it up along wavelength to produce spectra; they did this for several million objects. And the science folks have done with this data is astounding, from looking at distant quasars in the early universe, the cosmological distribution of galaxies and large scale structure, to the properties of stars in our own Galaxy. SDSS was truly revolutionary.

Every few years, SDSS gets upgraded. Maybe a specialized set of targets, maybe a new set of filters, maybe a new instrument. For instance, in 2005, SDSS started the aimed at spectroscopy of stars in our Galaxy to make a 3D map. In 2014, SDSS started the survey, a two-dimensional spectroscopy survey of nearly 10,000 nearby galaxies to study their star-forming regions. Most recently, SDSS was upgraded again, this time with the (see also ), another two-dimensional spectroscopic survey to target nearby galaxies to study their star-forming regions in detail.

That’s where I come in. I study emission lines for a living, particularly those produced in star-forming regions in other galaxies. Briefly, take the Orion Nebula for example, a star-forming region in our own Galaxy. It once was just a cloud of gas and dust, but eventually it collapsed under its own weight to form a new generation of stars. Of those newly formed stars, the hottest and most massive stars emit a lot of ultraviolet light that radiates outwards. Those photons of light hit any gas that has not yet formed stars, heating it up, exciting it, which causes the gas to emit its own light. We see that light as emission lines, lots of light at very particular wavelengths. For example, hydrogen has a series of emission lines in the optical called the Balmer lines: the brightest is called Hα (H-alpha), emitting at a wavelength of 6562 Å. The next brightest is Hβ (H-beta), emitting at a wavelength of 4861 Å. Other elements emit at other wavelengths dependent on their atomic structure. Singly-ionized oxygen (that is, oxygen with one electron removed) emits a lot of light at 3727 Å. Doubly-ionized oxygen (that is, oxygen with two electrons removed) emits two bright emission lines at 4959 Å and 5007 Å.

The visible hydrogen emission lines. Hα is the red line at the right, Hβ is the cyan line in the middle, Hγ is the next blue line, etc.

From these emission lines and their relative strengths (how bright they are compared with one another), we can learn a lot about the gas emitting them. We can learn about the abundance of elements in the gas, how highly ionized the gas is, the density of the gas, the temperature of the gas, so many things!

What’s the point of me telling you all of this? Although the LVM has started to collect data, they’ve only so far: the Helix Nebula (NGC 7293; see ). Unlike what I study, star-forming regions, the Helix Nebula is a planetary nebula in our own Galaxy. Planetary nebulae (which have nothing to do with planets or planet formation) are the sites of dead stars. When a star about the mass of our Sun dies, it will shed off the outer layers of its atmosphere, exposing its central hot core. We call that core a white dwarf. The white dwarf is very very very hot and emits a lot of ionizing light. That light will ionize what once was the star’s atmosphere and cause it to emit its own light, which we see as the nebula. Here’s the fun part: since there’s really only one star doing the ionizing, we’ll get a stratification of ionization: the highly ionized atoms and their light will be concentrated towards the center, with less ionized atoms and their light further out.

Alright, that’s enough introduction. Let’s see some data!

The Local Volume Mapper

The Local Volume Mapper instrument is what’s called an “integral field spectrograph” or IFS. These are fiber-fed instruments where each optic fiber collects light from a particular part of the sky. You can imagine it this way: for a given object, cover that object in small circles. Each circle is a fiber that is collecting the light only in that circle. The collection of those circles is the spectrograph and together we have not only spectroscopic information in single fibers, but spectroscopic information across a two-dimensional area. The plot below shows the fiber layout for this data of the Helix Nebula.

All of those circles are the individual fibers! Now, there are some holes. As the title of the plot suggests, I’ve only shown the “good” science fibers. The holes might be fibers that didn’t collect data, collected “bad” data (such as a star), or something was just wrong with the mechanical operation of that fiber. No worries, we’ve still got a lot of good fibers with good data.

Speaking of data, we can use this to extract spectral information about the Helix Nebula. For instance, if we select the region around the Hα line, we can make an “image” of the nebula at this wavelength.

Ta-da! The Helix Nebula! Compare this plot to the image at the beginning of this post from the Hubble Space Telescope. Can you see the similar structures?

Integrated Spectrum

One of the simplest things to do with an integral field spectrograph is to ignore the two-dimensional information and just add together the light from every fiber. This produces an “integrated” spectrum: the spectrum of the entire object. There’s a statistical benefit to doing this as well: since we’re adding up the light from a lot of fibers (1,754 fibers to be exact), the uncertainty in our measurements goes down. We’ll also be able to more clearly see faint emission lines, many of which are important to determine the physical quantities of the nebula. And we can see a lot of the lines! Not only are these observations from the LVM deep (meaning they stare at the object for a while), they sample a long wavelength range from the very blue optical light into the near-infrared (3600 - 9800 Å, in case you’re curious).

A quick note about emission line labeling. There are two types of emission line labeling, permitted and forbidden lines. Permitted lines are called permitted because they easily happen in labs here on Earth. In the plot below, these are lines without square brackets around them, such as the hydrogen lines Hα, Hβ, Hγ, etc. Forbidden lines are called forbidden because they don’t happen easily in labs on Earth; they require the extremely low densities of outer space to happen. In the plot below, these are the lines with square brackets around them. Take [O III] for instance. The “O” is the chemical symbol for that atom, in this case oxygen. But what are those Roman numerals doing there? That indicates the degree of ionization, i.e., how many electrons are missing. We call neutral “I”, so singly-ionized is “II”, doubly-ionized is “III”, triply-ionized is “IV”, etc. So “[O III]” tells us that this is a forbidden line of doubly-ionized oxygen! Similarly “[S II]” tells us that we have a forbidden line of singly-ionized sulfur. Finally, when referring to an emission line without the aid of a plot, we often write the wavelength of the line. So we might say “[O III]λ5007” to say that the wavelength of this line is at 5007 Å to distinguish it from another [O III] emission line.

Looking at the plot above, we see numerous strong and weak lines. (Note that I’ve cut off the $y$-axes; the strong lines extend above the plot.) Some of the strongest lines in nebulae are those of oxygen, hydrogen, nitrogen, and sulfur, such as [O II]λ3727, [O III]λλ4959,5007, Hβ, Hα, [N II]λλ6548,6584, and [S II]λλ6717,6731. See if you can identify them on the plot!

There are also a lot of faint lines. Some of these are very important for determining physical properties. Take for instance the faint [O III]λ4363 line right next to Hγ. It turns out that the ratio of this line to the stronger [O III]λ5007 line is completely dependent on the temperature of the highly-ionized gas, so if you can measure this ratio, then you know the temperature. There are other line ratios that help determine temperature too: for less ionized gas, you want to measure perhaps the ratio of the very blue [O II]λ3727 line and the very red [O II]λλ7320,7330 lines; or maybe the ratio of the strong [N II]λ6584 line to the weaker [N II]λ5755 line. All of these help determine the temperature of the gas.

Similarly, there are other lines that can measure the density of the gas. The sulfur lines are the most commonly used, [S II]λ6731/λ6717, since they are easily observable. Less-commonly used are the oxygen lines at 3727 Å. I’ve been labeling them as one line, but it’s actually a tightly-spaced doublet: one line at 3726 Å and another at 3729 Å. You need really finely-sampled measurements to resolve the individual lines. Thankfully, the LVM data is finely-sampled and we can resolve them!

However, it’s usually not as simple as measuring these line fluxes and dividing them. These two quantities, density and temperature, can depend on each other. So we have to use equations and techniques that take them both into account at the same time. For instance, let’s look at the [O III]λ5007/λ4363 line ratio. In the left plot, I’ve set the density to be 100 atoms/cm³, while in the right plot, I’ve set the temperature to be 10,000 K. But what happens when both vary together? We get something that looks like this:

Here, we’ve got two varying quantities: temperature on the vertical axis and density on the horizontal axis. Each of the colored lines and associated numbers are these values for a given line ratio. So when [O III]λ5007/λ4363 equals 3.4, you get the yellow line, for instance. Higher flux ratios are yellower colors, while lower flux rations are more purple colors. If you just look at the values at low densities (towards the left), the lines are flat. That means there’s no density-dependence, but as you go up the colors (i.e., the line ratios) change, indicating that there is a temperature-dependence. That’s exactly what we saw in the previous set of plots! However, as we go towards higher densities, the lines start to curve and even start to become vertical. We lose our temperature-dependence and start to gain a very strong density-dependence. How do we know which part of this plot our nebula is in?

Thankfully, each individual atom and their ions have their own versions of this complex plot. By detecting multiple emission lines, each with their own temperature- and density-dependence, we can narrow down on the exact temperature and density our nebula has! This is why we want tons of emission lines.

Measuring Emission Lines

But first, we have to measure the fluxes of these emission lines! To do that, we fit simple Gaussian functions to each of the emission lines. There are some technical details here that I’m not going to go into that are in the Notebook tutorial. I will mention one important thing: since we’re trying to fit multiple weak emission lines, in order to improve the fit we want to tie their wavelengths to how much they are separated from a strong line, like Hα. That’s exactly what I do in the Notebook tutorial. This drastically improves the fit and gives us very accurate fluxes. As an example, below is a fit to a strong set of lines, Hα+[N II]λλ6548,6584, and a weaker set of lines, Hγ+[O III]λ4363.

One final thing we have to do is correct our measured line fluxes for interstellar extinction. Space is full of dust. And that dust has two annoying effects: reddening and extinction. First, it generally dims light, and the light that it dims the most is blue light. You’re probably familiar with this if you’ve ever had the unfortunate experience of being near a wildfire and looked up at the Sun. Through the smoke, the Sun will appear dimmer and redder than it normally is. Thankfully, lots of very smart people have studied the extinction effects of dust in space and it is relatively simple nowadays to correct for this.

The most basic way to correct for dust is to look at the ratio of the hydrogen emission lines, usually Hα/Hβ. Quantum mechanics says that the ratio of these two lines should be equal to 2.86. This ratio does not have a strong dependence on temperature or density, so if our measured ratio is different from this, then it must be caused by interstellar dust. It turns out our measured value is about 2.94, so there does exist some dust but not a lot. We’ll calculate the amount of extinction happening and correct every line for it.

Estimating Physical Properties

Alright, now we can estimate some physical properties of our integrated spectrum! Our basic steps are to calculate the density first, then estimate some temperatures. Since we’re working with a planetary nebula and we expect some sort of stratification of ionization, we’ll need to calculate two different temperatures, one of the low-ionization ions and another for the high-ionization ions. Namely, we’ll use [S II]λλ6717,6731 to estimate densities, then use [O III]λλ4363,5007 for the high-ionization temperature and [O II]λλλ3727,7320,7330 for the low-ionization temperature.

Property	Value
Electron Density	45 cm^-3
Low-Ionization Temperature	13300 K
High-Ionization Temperature	9640 K

Looking at this table, these are pretty standard values for ionized gas. These agree with previous measurements as well: measure a density less than 100 cm^-3, a low-ionization temperature of 10,900 K, and a high-ionization temperature of 9,300 K. Our measurements are slighly higher for the temperatures, but this could be due to different measurement techniques and assumptions.

From those temperatures and densities, we can calculate the ionic abundance of each ion. This is the amount of each ion relative to hydrogen there are in the nebula. For historic reasons, we always set the amount of hydrogen equal to 10¹² (or 12 if we’re taking the logarithm), so every other ion will be less than this.

This will give us the abundances for the ions we observe, but what about the ions we don’t observe? For example, we observed [Ne III], but should we have observed [Ne II] or [Ne IV]? How much of these other ions should be present? Again, some really smart people have come up with equations called ionization correction factors or ICFs. These take our ionic abundances and correct them for the unseen ions so that we can calculate total abundances for each element, rather ionic abundances for each ion. Once we’ve picked a particular set of ICFs, we simply apply these and get back a total abundance for each element. Based on the emission lines we’ve detected, we’ll get total abundances for helium, oxygen, nitrogen, sulfur, argon, and neon.

Property	Value	Solar Value
Helium Abundance	11.16	10.93
Oxygen Abundance	8.67	8.69
Nitrogen Abundance	8.45	7.83
Sulfur Abundance	6.23	7.12
Neon Abundance	8.38	7.93
Argon Abundance	6.37	6.40

Here, I’m reporting abundances in that logarithmic form, where hydrogen is defined to have an abundance of 12; all other elements should ideally have an abundance less than 12. Looking at our table, the most abundant element is helium. That makes sense: helium is and stars will have produced a lot of helium in their lifetimes. After that comes oxygen, which also makes sense: oxygen is the third-most abundant element in the universe. Then we get the remaining elements, nitrogen, neon, argon, and sulfur, in order of decreasing abundance. Now we disagree with the universal abundances! What could be going on?

We have to remember that the elemental abundances here are the direct result of a star’s death, and . They are constantly changing light elements, like hydrogen and helium, into heavier elements through nuclear fusion. This could also explain the difference we see between the abundances of the Helix Nebula and the abundances found in the Sun. The Sun is only halfway through its evolution, and depending on the initial mass of the progenitor star of the Helix, it could have been massive enough to fuse heavier elements than the Sun will be able to. Indeed, those planetary nebulae with high abundances of helium and nitrogen are usually associated with high-mass progenitor stars. Granted, there are people who spend their whole careers studying these questions, but hopefully this gives you a little insight into why the Helix might have these abundances.

Two-Dimensional Spectra

All of that analysis was only for the integrated spectrum! However, we have a whole spatial dimension we haven’t done anything with yet. Now let’s make some maps of the different emission lines and estimate physical properties across the Helix Nebula. Thankfully, we’ve done most of the work already, we just have to apply our techniques to every individual fiber instead of the integrated spectrum. Of course, we’ll only want to plot fibers there is actual science data for, so we’ll ignore fibers that are staring at the background sky instead of the nebula.

One of our first outputs will be flux maps. These show the relative strengths of the different emission lines in each individual fiber. I’ve broken the images into the strong and weak lines.

Here we see different ionization morphologies, different shapes depending on the emission line. This is that idea of ionization stratification. The hydrogen lines, Hα and Hβ, essentially show the full extent of the ionized gas. The [O III]λ5007 emission line (labeled F_O3_5007 in the plot) is highly concentrated towards the center of the main ring. This is because it takes a lot of energy to doubly-ionize oxygen, so we need to be close to the central white dwarf to do that. We see similar, but fainter, concentration in some of the other doubly-ionized lines, like [S III]λλ9059,9531. The helium ion He IIλ4686 requires much more energy to be ionized, so it is much more centrally concentrated. In contrast, emission lines like [O II]λ3727, [N II]λλ6548,6584, and [S II]λλ6717,6731 highlight the low-ionization features, such as the faint outer “arm” that lies beyond the main ring.

Taken together, we have this picture that near the central white dwarf are very highly ionized ions like He II. As we move further away, we lose those highly energetic photons as they’re absorbed, resulting in this stratification. We go from He II, to [O III] and [S III], to [N II], [S II], and [O II], decreasing in ionization as we go. And even though they’re fainter, the weaker lines show this same stratification: [O III] at the center, then [Ne III] and [Ar III], then further out ions like [O II] and [N II].

Finally, we can make maps of the physical properties. I’ll focus on two maps, the density and the oxygen abundance.

The electron density map shows essentially the same purplish color throughout, indicating that most of the nebula is at the same density, approximately 100 atoms per cm³. Again, just like in our integrated case, this is pretty standard for ionized gas. The oxygen abundance map shows more interesting behavior. It looks like near the central star, there are lower abundances which then increase as we go further out. Is this a result of the changing temperatures across the region? Is there just less oxygen towards the center? Is it a mixture of both? I don’t really know. Just like before, there are many people who spend a lot of time thinking about these questions. I’ll point interested readers to the latter sections of for a good discussion.

The End

Well, that’s the end of this little project! I’ve learned a lot about planetary nebulae and direct-method measurements of elemental abundances. As part of this project, I’ve written a tutorial Notebook in case anyone wants to see what I’ve done. I did a lot more than what I’ve talked about here in this blog post. I’ve put that Notebook as well as a bunch more plots on my GitHub, which you can access using the link at the top of the page. As far as I know, the LVM team has only released one paper on the Helix Nebula as a proof of concept. Hopefully a future paper will be written doing a more in-depth and careful analysis of its abundances than I have done here. I’ll keep an eye out for it, and I’ll update this post with more info and comparisons. Hope you’ve enjoyed this detour into planetary nebulae abundances as much as I have!

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!