Harvard Astronomy 201b

Posts Tagged ‘dust’

ARTICLE: Interpreting Spectral Energy Distributions from Young Stellar Objects

In Journal Club, Journal Club 2013 on April 15, 2013 at 8:02 pm

Posted by: Meredith MacGregor

1. Introduction

When discussing young stellar objects and protoplanetary disks (as well as many other topics in astronomy), astronomers continually throw out the term ‘SED.  In early April, I attended a conference titled ‘Transformational Science with ALMA: From Dust to Rocks to Planets– Formation and Evolution of Planetary Systems.’  And, I can attest to the fact that the term ‘SED’ has come up in a very significant fraction of the contributed talks.  For those not intimately familiar with this sub-field, this rampant use of abbreviations can be confusing, making it difficult to glean any useful take-aways from a talk.  So, in addition to summarizing the article by Robitaille et al. (2007), the goal of this post is to give a bit of an introduction to the terminology and motivation for the growing field of star and planetary system formation.

SEDs: What They Are and Why We Care So Much

The abbreviation ‘SED’ stands for ‘Spectral Energy Distribution.’  If you want to sound like an expert, this should be pronounced exactly as it appears (analogous to the non-acronym ‘said’).  A SED is essentially a graph of flux versus wavelength.  In the context of the Robitaille et al. article, we are most interested in the SEDs for young stars and the envelopes and disks surrounding them.  So, why exactly does the flux from a young stellar object (YSO) vary with wavelength?  As it turns out, different regions of the YSO emit at different wavelengths.  This means that when we observe at different wavelengths, we are actually probing distinct regions of the star and its surrounding disk and envelope. By tracing out the entire SED for a YSO, we can determine what the geometry, structure, and constituents are for that object.

Before we go too far, it is worth taking a moment to clarify the terminology used to classify young stars.  A young stellar object or YSO is typically defined as any star in the earliest stages of development.  YSOs are almost always found in or near clouds of interstellar dust and gas.  The broad YSO class of objects is then divided into two sub-classes: protostars and pre-main sequence stars.  Protostars are heavily embedded in dust and gas and are thus invisible at optical wavelengths.  Astronomers typically use infrared, sub-millimeter, and millimeter telescopes to explore this stage of stellar evolution, where a star acquires the bulk of its material via infall and accretion of surrounding material.  Pre-main sequence (PMS) stars are defined to be low mass stars in the post-protostellar phase of evolution.  These stars have yet to enter the hydrogen-burning phase of their life and are sill surrounded by remnant accretion disks called ‘protoplanetary’ disks.  A more detailed a summary of this classification scheme can be found here.  However, because the evolution of young stars is far from well-understand, astronomers often typically these words interchangeably.

SEDs for pre-main sequence stars are often seen to have a bump or excess in the infrared. This bump is generally interpreted as being due to thermal emission from warm dust and gas surrounding the central star. As an illustrative example, let’s consider a protoplanetary disk around a young star. The image below is taken from a review article by Dullemond & Monnier (2010) and shows a graphical representation of the emission spectrum from such a disk.

dullemond

A graphical representation of the emission spectrum from a protoplanetary disk and the telescopes that can be used to probe different regions of the disk. Taken from Dullemond & Monnier (2010).

In this case, emission in the near-infrared traces the warmer inner regions of the disk.  As you move into submillimeter wavelengths, you start to probe the outer, cooler regions of the disk.  By modeling the observed SED of a pre-main sequence star, you can determine what components of the source are contributing to the observed flux.  The second figure below is taken from a paper by Guarcello et al. (2010).  The left panel of this figure shows the observed SED (solid line) of a BWE star (in line with other ridiculous astronomy acronyms, this stands from ‘blue stars with excesses’) and the unreddened photospheric flux we would expect to see if the star did not have a disk (dashed line).  The right panel shows a model fit to this data.  The authors describe the observed SED using a model with four distinct components, each represented by a colored line in the figure: (1) emission from the reddened photosphere of the central star, (2) radiation scattered into the line of sight from dust grains in the disk, (3) emission from a collapsing outer envelope, and (4) thermal emission from a circumstellar disk.  The summation of these four component makes up the complete SED for this BWE star.

guarcello

Left panel: The observed SED (solid line) of a BWE star and the unreddened photospheric flux we would expect to see if the star did not have a disk (dashed line). Right panel: A four component model fit to this data. Taken from Guarcello et al. (2010).

Describing and Classifying the Evolution of a Protostar

As a protostar evolves towards the Zero Age Main Sequence (ZAMS), the system geometry (and thus the SED) will evolve as well. Therefore, the stage of evolution of a protostar is often classified according to both the general shape and the features of the SED.  A graphical overview of the four stages of protostellar evolution are shown below (Andrea Isella’s thesis, 2006).  Class 0 objects are characterized by a very embedded central core in a much larger accreting envelope.  The mass of the central core grows in Class I objects and a flattened circumstellar accretion disk develops.  For Class II objects, the majority of circumstellar material is now found in a disk of gas and dust.  Finally, for Class III objects, the emission from the disk becomes negligible and the SED resembles a pure stellar photosphere.  The distinction between these different classes was initially defined by the slope of the SED (termed the ‘spectral index’) at infrared wavelengths.  Class I sources typically have SEDs that rise in the far- and mid-infrared, while Class II sources have flat or falling SEDs in the mid-infrared.  However, this quantitative distinction is not always clear and is not an effective way to unambiguously distinguish between the different object classes (Protostars and Planets V, page 127-128).

isella

A graphical overview of the four stages of protostar evolution taken from Andrea Isella’s thesis (2006). A typical SED of each class is shown in the left column and a cartoon of the corresponding geometry is shown in the right column.

2. The Article Itself

One common method of fitting SEDs is to assume a given gas and circumstellar dust geometry and set of dust properties (grain size distribution, composition, and opacity), and then use radiative transfer models to predict the resulting SED and find a set of parameters that best reproduce the observations.  However, fitting SEDs by trial and error is a time consuming way to explore a large parameter space.  The problem is even worse if you want to consider thousands of sources.  So, what’s to be done?  Enter Robitaille et al.  In order to attempt to make SED fitting more efficient, they have pre-calculated a large number of radiative transfer models that cover a reasonable amount of parameter space.  Then, for any given source, one can compare the observed SED to this set of models to quickly find the set of parameters that best explains the observations.

Let’s Get Technical

The online version of this fitting tool draws from 20,000 combinations of physical parameters and 10 viewing angles (if you are particularly curious, the online tool is available here).  A brief overview of the parameter space covered is as follows:

  • Stellar mass between 0.1 and 50 solar masses
  • Stellar ages between 10^3 and 10^7 years
  • Stellar radii and temperatures (derived directly from stellar mass using evolutionary tracks)
  • Disk parameters (disk mass, accretion rate, outer radius, inner radius, flaring power, and scale height) and envelope parameters (the envelope accretion rate, outer radius, inner radius, cavity opening angle, and cavity density) sampled randomly within ranges dictated by the age of the source

However, there are truly a vast number of parameters that could be varied in models of YSOs.  Thus, for simplicity, the authors are forced to make a number of assumptions.  Here are some of the biggest assumptions involved:

  1. All stars form via accretion through a disk and an envelope.
  2. The gas-to-dust ratio in the disk is 100.
  3. The apparent size of the source is not larger than the given aperture.

The last constraint is not required, but it allows a number of model SEDs to be cut out and thus speeds up the process.  Furthermore, the authors make a point of saying that the results can always be scaled to account for varying gas-to-dust ratios, since only the dust is taken into account in the actual radiative transfer calculations.

Does This Method Really Work?

If this tool works well, it should be able to correctly reproduce previous results.  In order to test this out, the authors turn to the Taurus-Auriga star forming region.  They select a sample of 30 sources from Kenyon & Hartman (1995) that are spatially resolved, meaning that there is prior knowledge of their evolutionary stage from direct observations (i.e. there is a known result that astronomers are fairly certain of to compare the model fits against).  When fitting their model SEDs with the observed SEDs for this particular star forming region, the authors throw in a few additional assumptions:

  1. All sources are within a distance range of 120 – 160 AU (helps to rule out models that are too faint or too luminous).
  2. The foreground interstellar extinction is no more than A_v = 20.
  3. None of the sources appeared larger than the apertures used to measure fluxes.

The authors then assign an arbitrary cut-off in chi-squared for acceptable model fits: \chi^2 - \chi^2_\text{best} < 3.  Here, \chi^2_\text{best} is the \chi^2 of the best-fit model for each source.  Robitaille et al. acknowledge that this cut-off has no statistical justification: ‘Athough this cut-off is arbitrary, it provides a range of acceptable fits to the eye.’  After taking Jim Moran’s Noise and Data Analysis class, I for one would like to see the authors try a Monte Carlo Markov Chain (MCMC) analysis of their 14-dimensional space (for more detail on MCMC methods see this review by Persi Diaconis).  That might make the analysis a bit less ‘by eye’ and more ‘by statistics.’

The upshot of this study is that for the vast majority of the sources considered, the best-fit values obtained by this new SED fitting tool are close to the previously known values. Check.

It is also worth mentioning here, that there are many other sets of SED modeling codes.  One set of codes of particular note are those written by Paola D’Alessio (D’Alessio et al., 1998; D’Alessio et al., 1999; D’Alessio et al, 2001).  These codes were the most frequently used in the results presented at the ALMA conference I attended.  The distinct change in the D’Alessio models is that they solve for the detailed hydrostatic vertical disk structure in order to account for observations of ‘flared’ disks around T Tauri stars (flaring refers to an increase in disk thickness at larger radii).

But, Wait! There are Caveats!

Although the overall conclusion is that this method fits SEDs with reasonable accuracy, there are a number of caveats that are raised.  First of all, the models tend to overestimate the mid-IR fluxes for DM Tau and GM Aur (two sources known to have inner regions cleared of dust).  The authors explain that this is most likely due to the fact that their models for central holes assume that there is no dust remaining in the hole.  In reality, there is most likely a small amount of dust that remains.  Second, the models do not currently account for the possibility of young binary systems and circumbinary disks (relevant for CoKu Tau 1).

The paper also addresses estimating parameters such as stellar temperature, disk mass, and accretion rate from SED fits.  And, yes, you guessed it, these calculations raise several more issues.  For very young objects, it is difficult to disentangle the envelope and disk, making it very challenging to estimate a total disk mass.  To make these complications clearer, the set of two plots below from the paper show calculated values for the disk mass plotted against the accepted values from the literature.  It is easily seen that the disk masses for the embedded sources are the most dissimilar from the literature values.

robitaille_disk_mass

Two plots from Robitaille et al. (2007) that show calculated values for the disk mass plotted against the accepted values from the literature. It is easily seen that the disk masses for the embedded sources (right) are the most dissimilar from the literature values.

Furthermore, even if the disk can be isolated, the dust mass in the disk is affected by the choice of dust opacity.  That’s a pretty big caveat!  A whole debate was started at the ALMA conference over exactly this issue and the authors have simply stated the problem and swept it under the rug in just one sentence.  In 2009, David Hogg conducted a survey of the rho Ophiucus region and used the models of Robitaille et al. (2007) to determine the best-fit dust opacity index, \beta for this group of sources.   Hogg found that \beta actually decreases for Class II protostars, a possible indication of the presence of larger grains in the disk.  Robitaille et al. also mention that the calculated accretion rates from SED fitting are systematically larger than what is presented in the literature.  The authors conclude that future models should include disk emission inside the dust destruction radius, the radius inside which it is too hot for dust to survive.  A great example of the complications that arise from a disk with a central hole can be seen in LkCa 15 (Espaillat et al., 2010Andrews et al., 2011).   The figure below shows the observed and simulated SEDs for the source (left) as well as the millimeter image (right).  The double ansae (bright peaks or ‘handles‘ apparent on either side of the disk) seen in the millimeter contours are indicative of a disk with a central cavity.

espaillat

Left: The observed and simulated SEDs for Lk Ca 15. The sharp peak seen at 10 microns is due to silicate grains within the inner hole of the disk. Right: The millimeter image of the disk. (Espaillat et al., 2010; Andrews et al., 2011)

In this case, a population of sub-micron sized dust within the hole is needed in order to produce the observed silicate feature at 10 microns.  Furthermore, an inner ring is required to produce the strong near-IR excess shortward of 10 microns.  A cartoon image of the predicted disk geometry is shown below.  To make things even more complicated, the flux at shorter wavelengths appears to vary inversely with the flux at longer wavelengths over time (Espaillat et al., 2010).  This phenomenon is explained by changing the height of the inner disk wall over time.

LkCa15

A cartoon image of the predicted disk geometry for Lk Ca 15 showing the outer ring, silicate grains within the hole, and the inner ring. (Espaillat et al., 2010)

Finally, Robitaille et al. discuss how well parameters are constrained given different combinations of data points for two example sources: AA Tau and IRAS 04361+2547.  In both sources, if only IRAC (Infrared Array Camera on the Spitzer Space Telescope) fluxes obtained between 3.6 and 8 microns are used, the stellar mass, stellar temperature, disk mass, disk accretion rate, and envelope accretion rate are all poorly constrained.  Things are particularly bad for AA Tau in this scenario, where only using IRAC data results in ~5% of all SED models meeting the imposed goodness of fit criterion (yikes!).  Adding in optical data to the mix helps to rule out models that have low central source temperatures and disk accretion rates.  Adding data at wavelengths longer than ~ 20 microns helps to constrain the evolutionary stage of the YSO, because that is where any infrared excess is most apparent.  And, adding submillimeter data helps to pin down the disk mass, since the emission at these longer wavelengths is dominated by the dust.  This just goes to show how necessary it is to obtain multi-wavelength data if we really want to understand YSOs, disks, and the like.

3. Sources:

Where to look if you want to read more about anything mentioned here…

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ARTICLE: The mass function of dense molecular cores and the origin of the IMF (2007)

In Journal Club, Journal Club 2013, Uncategorized on March 12, 2013 at 9:30 pm

The mass function of dense molecular cores and the origin of the IMF

by  Joao Alves Marco Lombardi & Charles Lada
Summary by Doug Ferrer

Introduction

One of the main goals of researching the ISM is understanding the connection between the number and properties of stars and the properties of the surrounding galaxy. We want to be able to look at a stellar population and deduce what sort of material it came from, and the reverse–predict what sort of stars we should expect to form in some region given what we know about its properties. The basics of this connection have been known for a while (eg. Bok 1977). Stars form from the gravitational collapse of dense cores of molecular clouds. Thus the properties of stars are the properties of these dense cores modulated by the physical processes that happen during this collapse.

One of the key items we would like to be able to derive from this understanding of star formation is the stellar initial mass function (IMF)–the number of stars of a particular mass as a function of mass. Understanding how the IMF varies from region to region and across time would be extremely useful in many areas of astrophysics, from cosmology to star clusters. In this paper, Alves et al. attempt to explain the origin of the IMF and provide evidence for this explanation by examining the molecular cloud complex in the Pipe nebula. We will look at some background on the IMF, then review the methods used by Alves et al. and asses the implications for star formation.

The IMF and its Origins

The IMF describes the probability density for any given star to have a mass M, or equivalently the number of stars in a given region with a mass of M. Early work done by Salpeter (1955) showed that the IMF for relatively high mass stars ( M > 1 M_{sun}) follows a power law with the index \alpha \approx -1.3 - 1.4. For lower masses, the current consensus is for a break below 1 M_{sun}, and another break below .1 M_{sun} , with a peak around .3 M_{sun}. A cartoon of this sort of IMF is shown in Fig. 1.  The actual underlying distribution may in fact be log-normal. This is consistent with stars being formed from collapsing over-densities caused by supersonic turbulence within molecular clouds (Krumholz 2011). This is not particularly strong evidence, however, as the log-normal distribution can result from any process that depends on the product of many random variables.

Example of the IMF

Figure 1: An illustration of the accepted form of the IMF. There is a power law region for less than 1 M_{sun}, and a broad peak for lower masses. Image from Krumholz (2011) produced using Chabrier (2003)

The important implication of these results is that there is a characteristic mass scale for star formation of around .2-.3 M_{sun} . There are two (obvious) ways to explain this:

  1. The efficiency of star formation peaks at certain scales.
  2. There is a characteristic scale for the clouds that stars form from

There has been quite a lot of theoretical work examining option 1 (see Krumhoz 2011 for a relatively recent, accessible review). There are many different physical processes at play in star formation–turbulent MHD, chemistry, thermodynamics, radiative transfer, and gravitational collapse.  Many of these processes are not separately well understood, and each is occurring in its complex, and highly non-linear regime.  We are not even close to a complete description of the full problem. Thus it would not be at all surprising if there were some mass scale, or even several mass scales that are singled out by the star formation process, even if none is presently known. There is some approximate analytical work showing that feedback from stellar outflows may provide such a scale (eg. Shu et al. 1987) . More recent work (e.g. Price and Bate 2008) has shown that magnetic fields cause significant effects on the structure of collapsing cloud cores in numerical simulations, and may reduce or enhance fragmentation depending on magnetic field strength and mass scale.

Nevertheless, the authors are skeptical of the idea that star-formation is not a scale-free process. Per Larson (2005), they do not believe that turbulence or magnetic fields are likely to be very important for the smallest, densest scale clouds that starts ultimately form from. Supersonic turbulence is quickly dissipated on these scales, and magnetic fields are dissipated through ambipolar diffusion–the decoupling of neutral molecules from the ionic plasma. Thus Larson argues that thermal support is the most important process in small cores, and the Jeans analysis will be approximately correct.

The authors thus turn to option 2. It is clear that if the dense cores of star forming clouds already follow a distribution like the IMF, then there will be no need for option 1 as an explanation. Unfortunately though, the molecular cloud mass function (Fig. 2) does not at first glance show any breaks at low mass and has too shallow of a power law index .  But what if we look at only the smallest, densest clumps?

Cloud mass Function

Figure 2: The cumulative cloud mass function (Ak is proortional to mass) for several different cloud complexes from Lombardi et al. (2008). While this is not directly comparable to the IMF, the important take away is that there are no breaks at low mass.

Observations

Observations using dense gas emission tracers like C^{18} O and HC^{13}O produces mass distributions more like the stellar IMF (eg. Tachihara et al. 2002, Onishi 2002) . However, there are many systematic uncertainties in emission based analysis. To deal with these issues, the authors instead probed dense cloud mass using wide field extinction mapping (this work).   An extinction map was constructed of the nearby Pipe nebula using the NICER method of Lombardi & Alves (2001), which we have discussed in class. This produced the extinction map shown in Fig. 3 below.

NICER Extinction map of the pipe nebula

Figure 3: NICER Extinction map of the Pipe nebula. A few dense regions are visible, but the noisey, variable background makes it difficult to seperate out seperate cores in a consistent way.

Data Analysis

The NICER map of the pipe nebula reveals a complex filamentary structure with very high dynamic range in column density (~ 10^{21} - 10^{22.5} cm^{-2}). It is difficult to assign cores to regions in such a data set in a coherent way (Alves et al. 2007) using standard techniques–how do we determine what is a core and what is substructure to a core? Since it is precisely the number and distribution of cores that we are interested in, we cannot use a biased method to identify the cores. To avoid this, the authors used a technique called  multiscale wavelet decomposition. Since the authors do not give much information on how this works, we will give a brief overview following a description of a similar technique from Portier-Fozzani et al. (2001).

Wavelet Decomposition

Wavelet analysis is a hybrid of Fourier and coordinate space techniques. A wavelet is a function that  has a characteristic frequency, position and spatial extent, like the one in Fig.  4. Thus if we convolve a wavelet of a given frequency and length with a signal, it will tell us how the spatial frequency of the signal varies with position. This is the type of analysis used to produce a spectrogram in audio visualization.

Wavelet example

Figure 4: A wavelet. This is one is the product of a sinusoid with a gausian envelope. This choice is a perfect tradeoff between spatial and frequency resolution, but other wavelets may be ideal for some other resolution goal. Note that \Delta x \Delta k \ge 1/2

The authors used this technique to separate out dense structures from the background. They performed the analysis using wavelets with spatial scales close to three different length scales separated by factors of 2 (.08 pc, .15 pc, .3 pc). Then they combined the structures identified at each length scale into trees based on spatial overlap, and called only the top level of these trees separate cores. The resulting identified cores are shown in fig. 5.  While the cores are shown as circles in fig. 5, this method does not assume spherical symmetry, and the actual identified core shapes are filamentary, reflecting the qualitative appearance of the image

Dense cores in pipe nebula

Figure 5: Dense cores identified in the Pipe nebula (circles) from Alves et al. (2007). The radii of the circles is proportional to the core radius determined by the wavelet decomposition analysis

Results

After obtaining what they claim to be a mostly complete sample of cores, the authors calculate the mass distribution for them. This is done by converting dust extinction to a dust column density. This gives a dust mass for each clump, which can then be extrapolated to a total mass by assuming a set dust fraction.  The result of this is shown in fig. 6 below. The core mass function they obtain is qualitatively similar in shape to the IMF, but scaled by a factor of ~4 in mass. The analysis is only qualitative, and no statistics are done or functions fit to the data. The authors claim that this result evinces a universal star formation efficiency of ~30%, and that this is good agreement with that calculated analytically by (Shu 2004) and numerical simulations. This is again only a qualitative similarity, however.

We should also note that the IMF is hypothesized to be a log-normal distribution. This sort of distribution can arise out of any process that depends multiplicatively on many independent random factors. Thus the fact that dense cores have a mass function that is a scaled version of the IMF is not necessarily good evidence that they share a simple causal link, in the same way that two variables both being normally distributed does not mean that they are any way related.

Figure 6: The mass function of dense molecular cores (points) and the IMF (solid grey line). The dotted gray line is the IMF with mass argument scaled by a factor of 4. The authors note the qualitative agreement, but do not perform any detailed analysis.

Conclusions

Based on this, the authors conclude that there is no present need for a favored mass scale in star formation as an explanation of the IMF.  Everything can be explained by the distribution produced by dense cloud cores as they are collapsing.  There are a few points of caution however. This is a study of only one cluster, and the data are analyzed using an opaque algorithm that is not publicly available. Additionally, the distributions are not compared statistically (such as with KS), so we have only qualitative similarity.  It would be interesting to see these results replicated for a different cluster using a more transparent statistical method

References

Bok, B. J., Sim, M. E., & Hawarden, T. G. 1977, Nature, 266, 145

Krumholz, M. R. 2011, American Institute of Physics Conference Series, 1386, 9

Lombardi, M., Lada, C. J., & Alves, J. 2008, A&A, 489, 143

Krumholz, M. R., & Tan, J. C. 2007, ApJ, 654, 304

Alves, J., Lombardi, M., & Lada, C. J. 2007, A&A, 462, L17

Lombardi, M., Alves, J., & Lada, C. J. 2006, A&A, 454, 781

Larson, R. B. 2005, MNRAS, 359, 211

Shu, F. H., Li, Z.-Y., & Allen, A. 2004, Star Formation in the Interstellar Medium: In Honor of
David Hollenbach, 323, 37

Onishi, T., Mizuno, A., Kawamura, A., Tachihara, K., & Fukui, Y. 2002, ApJ, 575, 950

Tachihara, K., Onishi, T., Mizuno, A., & Fukui, Y. 2002, A&A, 385, 909

Portier-Fozzani, F., Vandame, B., Bijaoui, A., Maucherat, A. J., & EIT Team 2001, Sol. Phys.,
201, 271

Shu, F. H., Adams, F. C., & Lizano, S. 1987, ARA&A, 25, 23

Salpeter, E. E. 1955, ApJ, 121, 161

CHAPTER: Introductory Remarks on Radiative Processes

In Book Chapter on February 28, 2013 at 3:10 am

(updated for 2013)


The goal of the next several sections is to build an understanding of how photons are produced by, are absorbed by, and interact with the ISM. We consider a system in which one or more constituents are excited under certain physical conditions to produce photons, then the photons pass through other constituents under other conditions, before finally being observed (and thus affected by the limitations and biases of the observational conditions and instruments) on Earth. Local thermodynamic equilibrium is often used to describe the conditions, but this does not always hold. Remember that our overall goal is to turn observations of the ISM into physics, and vice-versa.

The following contribute to an observed Spectral Energy Distribution:

      • gas: spontaneous emission, stimulated emission (e.g. masers), absorption, scattering processes involving photons + electrons or bound atoms/molecules
      • dust: absorption; scattering (the sum of these two -> extinction); emission (blackbody modified by wavelength-dependent emissivity)
      • other: synchrotron, brehmsstrahlung, etc.

The processes taking place in our “system” depend sensitively on the specific conditions of the ISM in question, but the following “rules of thumb” are worth remembering:

      1. Very rarely is a system actually in a true equilibrium state.
      2. Except in HII regions, transitions in the ISM are usually not electronic.
      3. The terms Upper Level and Lower Level refer to any two quantum mechanical states of an atom or molecule where E_{\rm upper}>E_{\rm lower}. We will use k to index the upper state, and j for the lower state.
      4. Transitions can be induced by photons, cosmic rays, collisions with atoms and molecules, and interactions with free electrons.
      5. Levels can refer to electronic, rotational, vibrational, spin, and magnetic states.
      6. To understand radiative processes in the ISM, we will generally need to know the chemical composition, ambient radiation field, and velocity distribution of each ISM component. We will almost always have to make simplifying assumptions about these conditions.

ARTICLE: Dark Nebulae, Globules, and Protostars

In Journal Club, Journal Club 2013 on February 19, 2013 at 10:45 pm

Dark Nebulae, Globules, and Protostars by Bart Bok (1977)

Summary by George Miller

Introduction

 

In Bart Bok’s 1977 paper Dark nebulae, globules, and protostars (Bok 1977), largely based on a lecture given upon acceptance of the Astroomical Society of the Pacific’s Bruce Medal, he presents two fundamentally different pictures of star formation. The first, constituting the majority of the paper’s discussion, occurs in large Bok globules which are compact, rounded and remarkably well-defined regions of high-extinction ranging from 3′ to 20′.  The globules show a strong molecular hydrogen and dust component and relatively little signs of higher neutral HI concentrations than its surroundings. In contrast, Bok briefly examines star formation in the Magellanic Clouds which show a vast amount of neutral atomic hydrogen and a comparatively small amount of cosmic dust. In this review, I will summarize a number of key points made by Bok, as well as provide additional information and modern developments since the paper’s original publishing.

Large Bok Globules

 

A history of observations

In 1908, Barnard drew attention to “a number of very black, small, sharply defined spots or holes” in observations of the emission nebula Messier 8 (Barnard 1908).  39 years later Bok published extensive observations of 16 “globules” present in M8 as well others in \eta Carinae, Sagittarius, Ophiuchus and elsewhere, making initial estimates of their distance, diameter and extinction (Bok & Reilly 1947). He further claimed that these newly coined “globules” were gravitationally contracting clouds present just prior to star formation, comparing them to an “insect’s cocoon” (Bok 1948). As we will see, this bold prediction was confirmed over 40 years later to be correct. Today there over 250 globules known within roughly 500 pc of our sun and, as Bok claims in his 1977 paper, identifying more distant sources is difficult due to their small angular diameter and large number of foreground stars.  There are currently four chief methods of measuring the column density within Bok Globules: extinction mappings of background stars, mm/sub-mm dust continuum emission, absorption measurements of galactic mid-IR background emission, and mapping molecular tracers.  See Figure 1 for a depiction of the first three of these methods.  At the time Bok published his paper in 1977, only extinction mapping and molecular tracer methods were readily available, thus I will primarily discuss these two.  For a more in depth discussion, see Goodman et. al. 2009 and the subsequent AST201b Journal Club review.

Figure 1.

Figure 1.  Three methods of determining column density of starless molecular cores or Bok globules. (a) K-band image of Barnard 68 and plot of the A_K as a function of radius from the core.  This method measures the H–K excess, uses the extinction law to convert into A_V, and then correlated to the H_2 column density from UV line measurements, parameterized by f. (b) 1.2-mm dust continuum emission map and flux versus radius for L1544.  \kappa_{\nu} is the dust opacity per unit gas mass, ρ is the dust density, and m the hydrogen mass (corrected for He). (c) 7-μm ISOCAM image and opacity versus radius for ρ Oph D.  In this method the absorbing opacity is related to the hydrogen column via the dust absorption cross section, \sigma_{\lambda}.  Figure taken from Bergin & Tafalla 2007.
 
 

Measuring photometric extinction

Measuring the photometric absorption, and thus yielding a minimum dust mass, for these globules is itself an arduous process. For globules with A_v<10   mag, optical observations with large telescopes can be used to penetrate through the globules and observe the background stars.  Here A_{\lambda} \equiv m_{\lambda}-m_{\lambda, 0} = 2.5 \, log(\frac{F_{\lambda,0}}{F_{\lambda}}).  Thus an extinction value of A_v=10 mag means the flux is decreased by a factor of 10^4.  By using proper statistics of the typical star counts and magnitudes seen within a nearby unobstructed field of view, extinction measurements can be made for various regions.  It is important to note that the smaller an area one tries to measure an extinction of, the greater the statistical error (due to a smaller number of background stars).  This is one of the key limitations of extinction mappings.  For the denser cores or more opaque globules with 10 < A_V < 20 mag, observations in the near infrared are needed (which is relatively simple by today’s standards but not so during Bok’s time). This is further complicated due to imprecisely defined BVRI photometric standard sequences for fainter stars, a problem still present today with various highly-sensitive space telescopes such as the HST. Bok mentions two methods. In the past a Racine (or Pickering) prism was used to produce fainter companion images of known standards, yet as discussed by Christian & Racine 1983 this method can produce important systematic errors. The second, and more widely used, method is to pick an easily accessible progression of faint stars and calibrate all subsequent photographic plates (or ccd images) from this. See Saha et. al. 2005 for a discussion of this problem in regards to the Hubble Space Telescope.

Obtaining an accurate photometric extinction for various regions within the globule is valuable as it leads an estimate of the dust density. Bok reports here from his previous Nature paper (Bok et. al. 1977) that the extinction A_v within the Coalsack Globule 2 varies inversely as the square of distance, thus also implying the dust density varies inversely as the cube of distance from the core.  Modern extinction mappings, as seen in Figure 1(a) of Barnard 68,  show that at as one approaches the central core the extinction vs. distance relation actually flattens out nearly to r^{-1}.  This result was a key discovery, for the Bonnor-Ebert (BE) isothermal sphere model predicts a softer power law at small radii.  In his paper, Bok remarks “The smooth density gradient seems to show that Globule 2 is […] an object that reminds one of the polytropic models of stars studied at the turn of the century by Lane (1870) and Emden (1907)”.  It is truly incredible how accurate this assessment was.  The Bonnor-Ebert sphere is a model derived from the Lane-Emden equation for an isothermal, self-gravitating sphere which remains in hydrostatic equilibrium.  Figure 2 displays a modern extinction mapping of Barnard 68 along with the corresponding BE sphere model, showing that the two agree remarkably well.  There are, however, a number of detractors from the BE model applied to Bok globules.  The most obvious is that globules are rarely spherical, implying that some other non-symmetric pressure must be present.  Furthermore, the density gradient between a globule’s core and outer regions often exceeds 14 (\xi_{max} > 6.5) as required for a stable BE sphere (Alves, Lada & Lada 2001).

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Figure 2.  Dust extinction mapping for Barnard 68 plotted against an isothermal Bonnor-Ebert sphere model.  Figure taken from Alves, Lada & Lada 2001.
 

Using CO as a tracer

Important tracer molecules, such as CO, are used to study the abundance of H_2, temperatures and kinematics of these globules. Because the more common ^{12}CO isotope tends to become optically thick and saturate in regions of higher column density such as globules, the strength of ^{13}CO emission is usually used to indicate the density of H2.  The conversion factor of N_{H_2} = 5.0 \pm 2.5 \times 10^5 \times N_{13}, from Dickman 1978 has changed little in over three decades. The column density of H_2, combined with its known mass and radius of the globule, can then be used to estimate the globule’s total mass. Furthermore, the correlation of ^{13}CO density with photometric extinction, A_v = 3.7 \times 10^{-16} \times N_{13}, is another strong indication that ^{13}CO emission is an accurate tracer for H_2 and dust. Further studies using C^{17}O and C^{18}O have also been used to trace even higher densities when even ^{13}CO can become optically thick(Frerking et. al. 1982).  As an example, Figure 3 shows molecular lines from the central region of the high-mass star forming region G24.78+0.08.  In the upper panel we can see the difference between the optically thick ^{12}CO and thin C^{18}O.  The ^{12}CO line shows obvious self-absorption peaks associated with an optically thick regime, and one clearly can not make a Gaussian fit to determine the line intensity.   ^{12}CO, due to the small dipole moment of its J=1 \rightarrow 0 transition and thus ability to thermalize at relatively low densities, is also used to measure the gas temperature within globules. These temperatures usually range from 7K to 15K. Finally, the width of CO lines are used to measure the velocity dispersion within the globule. As Bok states, most velocities range from 0.8 to 1.2 km/s. This motion is often complex and measured excess line-widths beyond their thermal values are usually attributed to turbulence (Bergin & Tafalla 2007). Importantly, the line-width vs. size relationship within molecular clouds first discovered by Barnard 1981 does not extend to their denser cores (which have similar velocity motions as Bok globules).  Instead, a “coherence” radius is seen where the non-thermal component of a linewidth is approximately constant (Goodman et. al. 1998).  In the end, as Bok surmises, the subsonic nature of this turbulence implies it plays a small role compared to thermal motions.

img32
 
Figure 3.  Spectra taken from the core of the high-mass star forming region G24.78+0.08.  The solid line corresponds to ^{12}CO (1\rightarrow 0), ^{12}CO (2\rightarrow 1), and C^{32}S (3\rightarrow 2), the dashed line to ^{13}CO (1\rightarrow 0)^{13}CO (2\rightarrow 1), and C^{34}S (3\rightarrow 2) and the dotted line to C^{18}O (1\rightarrow 0).  From the top panel, one can clearly see the difference between the optically thick, saturated ^{12}CO (1\rightarrow 0) line and the optically thin C^{18}O (1\rightarrow 0) transition.  Figure taken from Cesaroni et. al. 2003.
 
 

The current status of Bok globules

Today, the majority of stars are thought to originate within giant molecular clouds or larger dark cloud complexes, with only a few percent coming from Bok globules. However, the relative simplicity of these globules still make them important objects for studying star formation. While an intense debate rages today regarding the influence of turbulence, magnetic fields, and other factors on star formation in GMCs, these factors are far less important than simple gravitational contraction within Bok globules. The first list of candidate protostars within Bok globules, obtained by co-adding IRAS images, was published in 1990 with the apropos title “Star formation in small globules – Bart Bok was correct” (Yun & Clemens 1990).  To conduct the search, Yun & Clemens first fit a single-temperature modified blackbody model the the IRAS 60 and 100 μm images (after filtering out uncorrelated background emission) to obtain dust temperature and optical depth values.  This result was then used as a map to search for spatially correlated 12 and 25 μm point sources (see Figure 4.).  More evidence of protostar outflows (Yun & Clemens 1992), Herbig-Haro objects due to young-star jets (Reipurth et al. 1992) and the initial stages of protostar collapse (Zhou et. al. 1993) have also been detected within Bok Globules. Over 60 years after Bok’s pronouncement that these globules were “insect cocoons” encompassing the final stages of protostar formation, his hypothesis remains remarkably accurate and validated. It is truly “pleasant indeed that globules are there for all to observe!”

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Figure 4.    (a) Contour map of the dust temperature T_{60/100} of the Bok Globule CB60 derived from 60 and 100 μm IRAS images.  (b) 12 μm IRAS image of CB60 after subtracting background emission using median-filtering.  This source is thought to be a young stellar object or protostar located within the globule.  The other 12 μm field sources seen in (b) are thought not to be associated with the globule. Figure taken from Yun & Clemens 1990.
 

Magellanic Cloud Star Formation

 

At the end of his paper, Bok makes a 180 degree turn and discusses the presence of young stars and blue globulars within the Magellanic Clouds. These star formation regions stand in stark contrast to the previously discussed Bok globules; they contain a rich amount of HI and comparatively small traces of dust, they are far larger and more massive, and they form large clusters of stars as opposed to more isolated systems. Much more is known of the star-formation history in the MCs since Bok published this 1977 paper. The youngest star populations in the MCs are found in giant and supergiant shell structures which form filamentary structures throughout the cloud. These shells are thought to form from supernova, ionizing radiation and stellar wind from massive stars which is then swept into the cool, ambient molecular clouds. Further gravitational, thermal and fluid instabilities fragment and coalesce these shells into denser star-forming regions and lead to shell-shell interactions (Dawson et. al. 2013). The initial onset of this new (\sim 125 Myr) star formation is thought to be due to close encounters between the MCs, and is confirmed by large-scale kinematic models (Glatt et al. 2010).

References

Alves, J. F., Lada, C. J., & Lada, E. A. 2001, Nature, 409, 159
Barnard, E. E. 1908, Astronomische Nachrichten, 177, 231
Bergin, E. A., & Tafalla, M. 2007, ARA&A, 45, 339
Bok, B. J. 1948, Harvard Observatory Monographs, 7, 53
—. 1977, PASP, 89, 597
Bok, B. J., & Reilly, E. F. 1947, ApJ, 105, 255
Bok, B. J., Sim, M. E., & Hawarden, T. G. 1977, Nature, 266, 145
Cesaroni, R., Codella, C., Furuya, R. S., & Testi, L. 2003, A&A, 401, 227
Christian, C. A., & Racine, R. 1983, PASP, 95, 457
Dawson, J. R., McClure-Griffiths, N. M., Wong, T., et al. 2013, ApJ, 763, 56
Dickman, R. L. 1978, ApJS, 37, 407
Frerking, M. A., Langer, W. D., & Wilson, R. W. 1982, ApJ, 262, 590
Glatt, K., Grebel, E. K., & Koch, A. 2010, A&A, 517, A50
Goodman, A. A., Barranco, J. A., Wilner, D. J., & Heyer, M. H. 1998, ApJ, 504, 223
Goodman, A. A., Pineda, J. E., & Schnee, S. L. 2009, ApJ, 692, 91
Reipurth, B., Heathcote, S., & Vrba, F. 1992, A&A, 256, 225
Saha, A., Dolphin, A. E., Thim, F., & Whitmore, B. 2005, PASP, 117, 37
Yun, J. L., & Clemens, D. P. 1990, ApJ, 365, L73
—. 1992, ApJ, 385, L21
Zhou, S., Evans, II, N. J., Koempe, C., & Walmsley, C. M. 1993, ApJ, 404, 232

ARTICLE: On the Dark Markings in the Sky

In Journal Club, Journal Club 2013 on February 8, 2013 at 2:46 pm

On the Dark Markings in the Sky by Edward E. Barnard (1919)

Summary by Hope Chen

Abstract

By examining photographic plates of various regions on the sky, Edward E. Barnard concluded in this paper that what he called “dark markings” were in fact due to the obscuration of nearby nebulae in most cases. This result had a significant impact on the debate regarding the size and the dimension of the Milky Way and also the research of the interstellar medium, particularly work by Vesto Slipher, Heber Curtis and Robert Trumpler. The publication of  Photographic Atlas of Selected Regions of the Milky Way after Barnard’s death, which included many of the regions mentioned in the paper, further provided a new method of doing astronomy research. In this paper and the Atlas, we are also able to see a paradigm very different from that of today.

It is now well-known that the interstellar medium causes extinction of light from background stars. However, think of a time when the infrared imaging was impossible, and the word “photon” meant nothing but a suspicious idea. Back in such a time in the second decade of the twentieth century, Edward Edison Barnard, by looking at hundreds of photographic plates, proposed an insightful idea that “starless” patches of the sky were dark because they are obscured by nearby nebulae. This idea not only built the foundation of the modern concept of the interstellar medium, but also helped astronomers figure out that the Universe extended so much farther beyond the Milky Way.

Young Astronomer and His Obsession of the Sky

In 1919, E. E. Barnard published this paper and raised the idea that what he called “dark markings” are mostly obscuration from nebulae close to us. The journey, however, started long before the publication of this paper. Born in Nashville, Tennessee in 1857, Barnard was not able to receive much formal education owing to poverty. His first interest, which became important for his later career, was in photography. He started working as a photographer’s assistant at the age of nine, and the work continued throughout most of his teenage years. He then developed an interest in astronomy, or rather, “star-gazing,” and would go watch the sky almost every night with his own telescope. He took courses in natural sciences at Vanderbilt University and started his professional career as an astronomer at the Lick Observatory in 1888. He helped build the Bruce Photographic Telescope at the Lick Observatory and there he started taking pictures of the sky on photographic plates. He then moved on to his career at the Yerkes Observatory at Chicago University and worked there until his death in 1922. (Introduction of the Atlas, Ref. 2)

One of the many plates in the Atlas including the region around Rho Ophiuchii, which was constantly mentioned in many of Barnard's works.

Fig. 1 One of the many plates in the Atlas including the region around Rho Ophiuchii, which was constantly mentioned in many of Barnard’s works. (Ref. 2)

Fig. 1 is one of the many plates taken at the Yerkes Observatory. It shows the region near Rho Ophiuchii, which was a region constantly and repetitively visited by Barnard and his telescope. Barnard noted in his description of this plate, “the [luminous] nebula itself is a beautiful object. With its outlying connections and the dark spot in which it is placed and the vacant lanes running to the East from it, … it gives every evidence that it obscures the stars beyond it.” Numerous similar comments spread throughout his descriptions of various regions covered in A Photographic Atlas of Selected Regions of the Milky Way (hereafter, the Atlas). Then finally in his 1919 paper, he concluded, “To me these are all conclusive evidence that masses of obscuring matter exist in space and are readily shown on photographs with the ordinary portrait lenses,” although “what the nature of this matter may be is quite another thing.” The publication of these plates in the Atlas (unfortunately after his death, put together by Miss Mary R. Calvert, who was Barnard’s assistant at the Yerkes Observatory and helped publish many of Barnard’s works after his death) also provided a new way of conducting astronomical research just as the World Wide Telescope does today. The Atlas for the first time allowed researchers to examine the image and the astronomical coordinates along with lists of dominant objects at the same time.

Except quoting Vesto Slipher’s work on spectrometry measurements of these nebulae, most of the evidences in Barnard’s paper seemed rather qualitative than quantitative. So, as of today’s standard, was the “evidence” really conclusive? Again, the question cannot be answered without knowing the limits of astronomical research at the time. Besides an immature understanding of the underlying physics, astronomers in the beginning of the twentieth century were limited by the lack of tools on both the observation and analysis fronts. Photographic plates as those in the Atlas were pretty much the most advanced imaging technique at the time, on which even a quantitative description of “brightness” was not easy, not to mention an estimation of the extinction of these “dark markings.” However, this being said, a very meaningful and somewhat “quantitative” assumption was drawn in Barnard’s paper: the field stars were more or less uniformly distributed. Barnard came to this assumption by looking at many different places, both in the galactic plane and off the plane, and observing the densities of field stars in these regions. Although numbers were not given in the paper, this was inherently similar to a star count study. Eventually, this assumption lead to what Barnard thought as the conclusive evidence of these dark markings being obscuring nebulae instead of “vacancies.” Considering the many technical limits at the time, while the paper might not seem to be scientific in today’s standard, this paper did pose a “conclusion” which was strong enough to sustain many of the more quantitative following examinations.

The “Great Debate”

Almost at the same time, perviously mentioned Vesto Slipher (working at the Lowell Observatory) began taking spectroscopic measurements of various clouds and tried to understand the constituents of these clouds. Although limited by the wavelength range and the knowledge of different radiative processes (the molecular transition line emission used largely in the research of the interstellar medium today was not observed until half a century later in 1970, by Robert Wilson, who, on a side note, also discovered the Cosmic Microwave Background), Slipher was able to determine the velocities of clusters by measuring the Doppler shifts and concluded that many of these clusters move at a faster rate than the escape velocity of the Milky Way (Fig. 2). This result, coupled with Barnard’s view of intervening nebulae, revolutionized the notion of the Universe in the 1920s.

The velocity measurements from spectroscopic observations done by Vesto Slipher.

Fig. 2 The velocity measurements from spectroscopic observations done by Vesto Slipher. (Ref. 3)

On April 26, 1920 (and in much of the 1920s), the “Great Debate” took place between Harlow Shapley (the Director of Harvard College Observatory at the time) and Curtis Heber (the Lick Observatory, 1902 to 1920). The general debate concerned the dimension of the Universe and the Milky Way, but the basic issue was simply whether distant “spiral nebulae” were small and lay within the Milky Way or whether they were large and independent galaxies. Besides the distance and the velocity measurements, which suffered from large uncertainties due to the technique available at the time, Curtis Heber was able to “win” the debate by claiming that dark lanes in the “Great Andromeda Nebula” resemble local dark clouds as those observed by Barnard (Fig. 3, taken in 1899). The result of the debate then sparked a large amount of work on “extragalactic astronomy” in the next two decades and was treated as the beginning of this particular research field.

The photographic plate of the "Great Andromeda Nebula" taken in 1988 by Isaac Roberts.

Fig. 3 The photographic plate of the “Great Andromeda Nebula” taken in 1988 by Isaac Roberts.

The Paper Finally Has a Plot

Then after the first three decades of the twentieth century, astronomers were finally equipped with a relatively more correct view of the Universe, the idea of photons and quantum theory. In 1930, Robert J. Trumpler (the namesake of the Trumpler Award) published his paper about reddening and reconfirmed the existence of local “dark nebulae.” Fig. 4 shows the famous plot in his paper which showed discrepancies between diameter distances and photometric distances of clusters. In the same paper, Trumpler also tried to categorize effects of the ISM on light from background stars, including what he called “selective absorption” or reddening as it is known today. This paper, together with many of Trumpler’s other papers, is one of the first systematic research results in understanding the properties of Barnard’s dark nebulae, which are now known under various names such as clouds, clumps, and filaments, in the interstellar medium.

Trumpler's measurements of diameter distances v. photometric distances for various clusters.

Fig. 4 Trumpler’s measurements of diameter distances v. photometric distances for various clusters.

Moral of the Story

As Alyssa said in class, it is often more beneficial than we thought to understand what astronomers knew and didn’t know at different periods of time and how we came to know what we see as common sense today, not only in the historically interesting sense but also in the sense of better understanding of various ideas. In this paper, Barnard demonstrated a paradigm which we may call unscientific today but made a huge leap into what later became the modern research field of the interstellar medium.

Selected References

  1. On the Dark Markings in the Sky, E. E. Barnard (1919)
  2. A Photographic Atlas of Selected Regions of the Milky Way, E. E. Barnard, compiled by Edwin B. Frost and Mary R. Calvert (1927)
  3. Spectrographic Observations of Nebulae, V. M. Slipher (1915)
  4. Absorption of Light in the Galactic System, R. J. Trumpler (1930)

CHAPTER: Bruce Draine’s List of Constituents of the ISM

In Book Chapter on February 5, 2013 at 9:09 pm

(updated for 2013)

  1. Gas
  2. Dust
  3. Cosmic Rays*
  4. Photons**
  5. B-Field
  6. Gravitational Field
  7. Dark Matter

*cosmic rays are highly relativistic, super-energetic ions and electrons

**photons include:

  • The Cosmic Microwave Background (2.7 K)
  • starlight from stellar photospheres (UV, optical, NIR,…)
  • h\nu from transitions in atoms, ions, and molecules
  • “thermal emission” from dust (heated by starlight, AGN)
  • free-free emission (bremsstrahlung) in plasma
  • synchrotron radiation from relativistic electrons
  • \gamma-rays from nuclear transitions

His list of “phases” from Table 1.3:

  1. Coronal gas (Hot Ionized Medium, or “HIM”): T> 10^{5.5}~{\rm K}. Shock-heated from supernovae. Fills half the volume of the galaxy, and cools in about 1 Myr.
  2. HII gas: Ionized mostly by O and early B stars. Called an “HII region” when confined by a molecular cloud, otherwise called “diffuse HII”.
  3. Warm HI (Warm Neutral Medium, or “WNM”): atomic, T \sim 10^{3.7}~{\rm K}. n\sim 0.6 ~{\rm cm}^{-3}. Heated by starlight, photoelectric effect, and cosmic rays. Fills ~40% of the volume.
  4. Cool HI (Cold Neutral Medium, or “CNM”). T \sim 100~{\rm K}, n \sim 30 ~{\rm cm}^{-3}. Fills ~1% of the volume.
  5. Diffuse molecular gas. Where HI self-shields from UV radiation to allow H_2 formation on the surfaces of dust grains in cloud interiors. This occurs at 10~{\rm to}~50~{\rm cm}^{-3}.
  6. Dense Molecular gas. “Bound” according to Draine (though maybe not). n >\sim 10^3 ~{\rm cm}^{-3}. Sites of star formation.  See also Bok Globules (JC 2013).
  7. Stellar Outflows. T=50-1000 {\rm K}, n \sim 1-10^6 ~{\rm cm}^{-3}. Winds from cool stars.

These phases are fluid and dynamic, and change on a variety of time and spatial scales. Examples include growth of an HII region, evaporation of molecular clouds, the interface between the ISM and IGM, cooling of supernova remnants, mixing, recombination, etc.

CHAPTER: Composition of the ISM

In Book Chapter on February 5, 2013 at 9:03 pm

(updated for 2013)

  • Gas: by mass, gas is 60% Hydrogen, 30% Helium. By number, gas is 88% H, 10% He, and 2% heavier elements
  • Dust: The term “dust” applies roughly to any molecule too big to name. The size distribution is biased towards small (0.2 \mum) particles, with an approximate distribution N(a) \propto a^{-3.5}. The density of dust in the galaxy is \rho_{\rm dust} \sim .002 M_\odot ~{\rm pc}^{-3} \sim 0.1 \rho_{\rm gas}
  • Cosmic Rays: Charged, high-energy (anti)protons, nuclei, electrons, and positrons. Cosmic rays have an energy density of 0.5 ~{\rm eV ~ cm}^{-3}. The equivalent mass density (using E = mc^2) is 9 \times 10^{-34}~{\rm g cm}^{-3}
  • Magnetic Fields: Typical field strengths in the MW are 1 \mu G \sim 0.2 ~{eV ~cm}^{-3}. This is strong enough to confine cosmic rays.