Star Formation

Mark R Krumholz

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789813142053
• Topic: Sciences physiques
• Subtopic: Astronomie et astrophysique

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This book provides a modern introduction to the study of star formation, at a level suitable for graduate students or advanced undergraduates in astrophysics. The first third of the book provides a review of the observational phenomenology and then the basic physical processes that are important for star formation. The remainder then discusses the major observational results and theoretical models for star formation on scales from galactic down to planetary. The book includes recommendations for complementary reading from the research literature, as well as five problem sets with solutions.

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Contents:

• Introduction and Phenomenology:
  • Observing the Cold Interstellar Medium
  • Observing Young Stars
• Physical Processes:
  • Chemistry and Thermodynamics
  • Gas Flows and Turbulence
  • Magnetic Fields and Magnetized Turbulence
  • Gravitational Instability and Collapse
  • Stellar Feedback
• Star Formation Processes and Problems:
  • Giant Molecular Clouds
  • The Star Formation Rate at Galactic Scales: Observations
  • The Star Formation Rate at Galactic Scales: Theory
  • Stellar Clustering
  • The Initial Mass Function: Observations
  • The Initial Mass Function: Theory
  • Protostellar Disks and Outflows: Observations
  • Protostellar Disks and Outflows: Theory
  • Protostar Formation
  • Protostellar Evolution
  • Massive Star Formation
  • The First Stars
  • Late-Stage Stars and Disks
  • The Transition to Planet Formation

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Readership: Undergraduate and graduate students as well as researchers interested in astrophysics.
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Part III

Star Formation Processes and Problems

8

Giant Molecular Clouds

Suggested background reading:

•Dobbs, C. L., et al. 2014, in “Protostars and Planets VI”, ed. H. Beuther et al., pp. 3–26.

Suggested literature:

•Colombo, D., et al. 2014, ApJ, 784, 3.

We now begin our top–down study of star formation, from large to small scales. This chapter focuses on observations of the bulk properties of giant molecular clouds (GMCs), primarily in the Milky Way and in nearby galaxies where we can resolve individual GMCs. The advantage of looking at the Milky Way is of course higher resolution. The advantage of looking at other galaxies is that, unlike in the Milky Way, we can get an unbiased view of all the GMCs, with much smaller distance uncertainties and many fewer confusion problems. This allows us to make statistical inferences that are often impossible to check with confidence locally. This study will be a preparation for the next two chapters, which discuss the correlation of molecular clouds with star formation and the problem of the star formation rate.

8.1Molecular Cloud Masses

8.1.1Mass Measurement

The most basic quantity we can measure for a molecular cloud is its mass. However, this also turns out to be one of the trickiest quantities to measure. The most commonly used method for inferring masses is based on molecular line emission, because lines are bright and easy to see even in external galaxies. The three most commonly-used species on the galactic scale are ¹²CO, ¹³CO, and, more recently, HCN.

Optically Thin Lines: Conceptually, ¹³CO is the simplest, because its lines are generally optically thin. For emitting molecules in local thermodynamic equilibrium (LTE) at temperature T, it is easy to show from the radiative transfer equation that the intensity emitted by a cloud of optical depth τ_ν at frequency ν is simply

where B_ν(T) is the Planck function evaluated at frequency ν and temperature T.

Although we will not derive this equation here¹, it behaves exactly as one would expect intuitively. In the limit of a very optically thick cloud, τ_ν ≫ 1, the exponential factor becomes zero, and the intensity simply approaches the Planck function, which is the intensity emitted by a black body. In the limit of a very optically thin cloud, τ_ν ≪ 1, the exponential factor just becomes 1 − τ_ν, so the intensity approaches that of a black body multiplied by the (small) optical depth. Thus, the intensity is simply proportional to the optical depth, which is proportional to the number of atoms along the line of sight.

These equations allow the following simple method of deducing the column density from an observation of the ¹³CO and ¹²CO J = 1 → 0 lines (or any similar pair of J lines) from a molecular cloud. If we assume that the ¹²CO line is optically thick, as is almost always the case, then we can approximate 1 − e^−τ_ν ≈ 1 at line center, so I_ν ≈ B_ν (T). If we measure I_ν, we can therefore immediately deduce the temperature T. We then assume that the ¹³CO molecules are at the same temperature, so that B_ν(T) is the same for ¹²CO and ¹³CO except for the slight shift in frequency. Then if we measure I_ν for the center of the ¹³CO line, we can solve the equation

for τ_ν, the optical depth of the ¹³CO line. If N_13CO is the column density of ¹³CO atoms, then for gas in LTE the column densities of atoms in the level 0 and 1 states are

where Z is the partition function, which is a known function of T, and T₁ = 5.3 K is the temperature corresponding to the first excited state.

The opacity to line absorption at frequency ν is

where B₀₁ and B₁₀ are the Einstein coefficients for spontaneous absorption and stimulated emission, defined by

The quantity ϕ(ν) is the line shape function (see Chapter 1). The corresponding optical depth at line center is

Since we know t_ν from the line intensity, we can measure ϕ(ν) just by measuring the shape of the line, and N₀ and N₁ depend only on

and the (known) temperature, we can solve for

. In practice, we generally do this in a slightly more sophisticated way, by fitting the optical depth and line shape as a function of frequency simultaneously, but the idea is the same. We can then convert to an H₂ column density by assuming a ratio of ¹²CO to H₂, and of ¹³CO to ¹²CO.

This method also has some significant drawbacks that are worth mentioning. The need to assume ratios of ¹³CO to ¹²CO and ¹²CO to H₂ are obvious ones. The former is particularly tricky, because there is strong observational evidence that the ¹³C to ¹²C ratio varies with galactocentric radius. We also need to assume that the ¹²CO and ¹³CO molecules are at the same temperature, which may not be true because the ¹²CO emission comes mostly from the cloud surface and the ¹³CO comes from the entire cloud. Since the cloud surface is usually warmer than its deep interior, this will tend to make us overestimate the excitation temperature of the ¹³CO molecules, and thus underestimate the true column density. This problem can be even worse because the lower abundance of ¹³CO means that it cannot self-shield against dissociation by interstellar UV light as effectively at ¹²CO. As a result, it may simply not be present in the outer parts of clouds at all, leading us to miss their mass and underestimate the true column density.

Another serious worry is the assumption that the ¹³CO molecules are in LTE. As shown in Problem Set 1, the ¹²CO J = 1 state has a critical density of a few thousand cm⁻³, which is somewhat above the mean density in a GMC even when we take into account the effects of turbulence driving mass to high density. The critical density for the ¹³CO J = 1 state is similar. For the ¹²CO J = 1 state, the effective critical density is lowered by optical depth effects, which thermalize the low-lying states. Since ¹³CO is optically thin, however, there is no corresponding thermalization for it, so in reality the excitation of the gas tends to be sub-LTE. The result is that the emission is less than we would expect based on an LTE assumption, and so we tend to underestimate the true ¹³CO column density, and thus the mass, using this method.

A final point to mention about this method is that, since the ¹³CO line is optically thin, it is simply not as bright as an optically thick line would be. Consequently, this method is generally only used within the Galaxy, not for external galaxies.

Optically Thick Lines: Optically thick lines are nice and bright, so we can see them in distant galaxies. The challenge for an optically thick line is how to infer a mass, given that we are really only seeing the surface of a cloud. Our standard approach here is to define an “X factor”: a scaling between the observed frequency-integrated intensity along a given line of sight and the column density of gas along that line of sight. For example, if we see a frequency-integrated CO J = 1 → 0 intensity I_CO along a given line of sight, we define X_CO = N/I_CO, where N is the t...