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CHAPTER: The Virial Theorem

In Book Chapter on March 12, 2013 at 3:21 pm

(Transcribed by Bence Beky). See also these slides from lecture

See Draine pp 395-396 and appendix J for more details.

The Virial Theorem provides insight about how a volume of gas subject to many forces will evolve. Lets start with virial equilibrium. For a surface S,

$0 = \frac12 \frac{\mathrm D^2I}{\mathrm Dt^2} = 2\Gamma + 3\Pi + \mathscr M + W + \frac1{4\pi}\int_S(\mathbf r \cdot \mathbf B_ \mathbf B \cdot \mathrm d \mathbf s - \int_S \left(p+\frac{B^2}{8\pi}\right)\mathbf r \cdot \mathrm d\mathbf s,$

see Spitzer pp.~217–218. Here I is the moment of inertia:

$I = \int \varrho r^2 \mathrm dV$

$\Gamma$ is the bulk kinetic energy of the fluid (macroscopic kinetic energy):

$\Gamma = \frac12 \int \varrho v^2 \mathrm dV,$

$\Pi$ is $\frac23$ of the random kinetic energy of thermal particles (molecular motion), or $\frac13$ of random kinetic energy of relativistic particles (microscopic kinetic energy):

$\Pi = \int p \mathrm dV,$

$\mathscr M$ is the magnetic energy within S:

$\mathscr M = \frac1{8\pi} \int B^2 \mathrm dV$

and W is the total gravitational energy of the system if masses outside S don’t contribute to the potential:

$W = - \int \varrho \mathbf r \cdot \nabla \Phi \mathrm dV.$

Among all these terms, the most used ones are $\Gamma$ , $\mathscr M$ and $W$ . But most often the equation is just quoted as $2\Gamma+W=0$ . Note that the virial theorem always holds, inapplicability is only a problem when important terms are omitted.

This kind of simple analysis is often used to determine how bound a system is, and predict its future, e.g. collapse, expansion or evaporation. Specific examples will show up later in the course, including instability analyses.

The virial theorem as Chandrasekhar and Fermi formulated it in 1953 is the following:

$\underbrace {2T_m} _{2\Gamma} + \underbrace {2T_k} _{3\Pi} + \underbrace {\Omega} _{W} + \mathscr M = \underbrace {0} _{\frac {\mathrm D^2 I} {\mathrm D t^2}}.$

This uses a different notation but expresses the same idea, which is very useful in terms of the ISM.

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excitation and ionization, non-thermal processes, photons and radiation, thermodynamics

CHAPTER: Ion-Neutral Reactions

In Book Chapter on March 7, 2013 at 3:20 pm

(updated for 2013)

In Ion-Neutral reactions, the neutral atom is polarized by the electric field of the ion, so that interaction potential is

$U(r) \approx \vec{E} \cdot \vec{p} = \frac{Z e} {r^2} ( \alpha \frac{Z e}{r^2} ) = \alpha \frac{Z^2 e^2}{r^4}$ ,

where $\vec{E}$ is the electric field due to the charged particle, $\vec{p}$ is the induced dipole moment in the neutral particle (determined by quantum mechanics), and $\alpha$ is the polarizability, which defines $\vec{p}=\alpha \vec{E}$ for a neutral atom in a uniform static electric field. See Draine, section 2.4 for more details.

This interaction can take strong or weak forms. We distinguish between the two cases by considering b, the impact parameter. Recall that the reduced mass of a 2-body system is $\mu' = m_1 m_2 / (m_1 + m_2)$ In the weak regime, the interaction energy is much smaller than the kinetic energy of the reduced mass:

$\frac{\alpha Z^2 e^2}{b^4} \ll\frac{\mu' v^2}{2}$ .

In the strong regime, the opposite holds:

$\frac{\alpha Z^2 e^2}{b^4} \gg\frac{\mu' v^2}{2}$ .

The spatial scale which separates these two regimes corresponds to $b_{\rm crit}$ , the critical impact parameter. Setting the two sides equal, we see that $b_{\rm crit} = \big(\frac{2 \alpha Z^2 e^2}{\mu' v^2}\big)^{1/4}$

The effective cross section for ion-neutral interactions is

$\sigma_{ni} \approx \pi b_{\rm crit}^2 = \pi Z e (\frac{2 \alpha}{\mu'})^{1/2} (\frac{1}{v})$

Deriving an interaction rate is tricker than for neutral-neutral collisions because $n_i \ne n_n$ in general. So, let’s leave out an explicit n and calculate a rate coefficient k instead, in ${\rm cm}^3 {\rm s}^{-1}$ .

$k = <\sigma_{ni} v>$ (although really $\sigma_{ni} \propto 1/v$ , so k is largely independent of v). Combining with the equation above, we get the ion-neutral scattering rate coefficient

$k = \pi Z e (\frac{2 \alpha}{\mu'})^{1/2}$

As an example, for $C^+ - H$ interactions we get $k \approx 2 \times 10^{-9} {\rm cm^{3} s^{-1}}$ . This is about the rate for most ion-neutral exothermic reactions. This gives us

$\frac{{\rm rate}}{{\rm volume}} = n_i n_n k$ .

So, if $n_i = n_n = 1$ , the average time $\tau$ between collisions is 16 years. Recall that, for neutral-neutral collisions in the diffuse ISM, we had $\tau \sim 500$ years. Ion-neutral collisions are much more frequent in most parts of the ISM due to the larger interaction cross section.

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excitation and ionization, non-thermal processes, photons and radiation, thermodynamics

CHAPTER: Neutral-Neutral Interactions

In Book Chapter on March 7, 2013 at 3:19 pm

(updated for 2013)

Short range forces involving “neutral” particles (neutral-ion, neutral-neutral) are inherently quantum-mechanical. Neutral-neutral interactions are very weak until electron clouds overlap ( $\sim 1 \AA\sim 10^{-8}$ cm). We can therefore treat these particles as hard spheres. The collisional cross section for two species is a circle of radius r1 + r2, since that is the closest two particles can get without touching.

$\sigma_{nn} \sim \pi (r_1 + r_2)^2 \sim 10^{-15}~{\rm cm}^2$

What does that collision rate imply? Consider the mean free path:

$mfp = \ell_c \approx (n_n \sigma_{nn})^{-1} = \frac{10^{15}} {n_H}~{\rm cm}$

This is about 100 AU in typical ISM conditions ( $n_H = 1 {\rm cm^{-3}}$ )

In gas at temperature T, the mean particle velocity is given by the 3-d kinetic energy: $3/2 m_n v^2 = kT$ , or

$v = \sqrt{\frac{2}{3} \frac{kT}{m_n}}$ , where $m_n$ is the mass of the neutral particle. The mean free path and velocity allows us to define a collision timescale:

$\tau_{nn} \sim \frac{l_c}{v} \sim (\frac{2}{3} \frac{kT}{m_n})^{-1/2} (n_n \sigma_{nn})^{-1} = 4.5 \times 10^3~n_n^{-1}~T^{-1/2}~{\rm years}$ .

For (n,T) = ( $1~{\rm cm^{-3}, 80~K}$ ), the collision time is 500 years
For (n,T) = ( $10^4~{\rm cm^{-3}, 10~K}$ ), the collision time is 1.7 months
For (n,T) = ( $1~{\rm cm^{-3}, 10^4~K}$ ), the collision time is 45 years

So we see that density matters much more than temperature in determining the frequency of neutral-neutral collisions.

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atomic and molecular gas, chemistry, emission/absorption of spectral lines, excitation and ionization, photons and radiation, thermodynamics

CHAPTER: Excitation Processes: Collisions

In Book Chapter on March 7, 2013 at 3:18 pm

(updated for 2013)

Collisional coupling means that the gas can be treated in the fluid approximation, i.e. we can treat the system on a macrophysical level.

Collisions are of key importance in the ISM:

cause most of the excitation
can cause recombinations (electron + ion)
lead to chemical reactions

Three types of collisions

Coulomb force-dominated ( $r^{-1}$ potential): electron-ion, electron-electron, ion-ion
Ion-neutral: induced dipole in neutral atom leads to $r^{-4}$ potential; e.g. electron-neutral scattering
neutral-neutral: van der Waals forces -> $r^{-6}$ potential; very low cross-section

We will discuss (3) and (2) below; for ion-electron and ion-ion collisions, see Draine Ch. 2.

In general, we will parametrize the interaction rate between two bodies A and B as follows:

${\frac{\rm{reaction~rate}}{\rm{volume}}} = <\sigma v>_{AB} n_a n_B$

In this equation, $<\sigma v>_{AB}$ is the collision rate coefficient in $\rm{cm}^3 \rm{s}^{-1}. <\sigma v>_{AB}= \int_0^\infty \sigma_{AB}(v) f_v~dv$ , where $\sigma_{AB} (v)$ is the velocity-dependent cross section and $f_v~dv$ is the particle velocity distribution, i.e. the probability that the relative speed between A and B is v. For the Maxwellian velocity distribution,

$f_v~dv = 4 \pi \left(\frac{\mu'}{2\pi k T}\right)^{3/2} e^{-\mu' v^2/2kT} v^2~dv$ ,

where $\mu'=m_A m_B/(m_A+m_B)$ is the reduced mass. The center of mass energy is $E=1/2 \mu' v^2$ , and the distribution can just as well be written in terms of the energy distribution of particles, $f_E dE$ . Since $f_E dE = f_v dv$ , we can rewrite the collision rate coefficient in terms of energy as

$\sigma_{AB}=\left(\frac{8kT}{\pi\mu'}\right)^{1/2} \int_0^\infty \sigma_{AB}(E) \left(\frac{E}{kT}\right) e^{-E/kT} \frac{dE}{kT}$ .

These collision coefficients can occasionally be calculated analytically (via classical or quantum mechanics), and can in other situations be measured in the lab. The collision coefficients often depend on temperature. For practical purposes, many databases tabulate collision rates for different molecules and temperatures (e.g., the LAMBDA databsase).

For more details, see Draine, Chapter 2. In particular, he discusses 3-body collisions relevant at high densities.

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equilibria, excitation and ionization, photons and radiation, thermal (blackbody) emission, thermodynamics

CHAPTER: Definitions of Temperature

In Book Chapter on March 7, 2013 at 3:27 am

(updated for 2013)

The term “temperature” describes several different quantities in the ISM, and in observational astronomy. Only under idealized conditions (i.e. thermodynamic equilibrium, the Rayleigh Jeans regime, etc.) are (some of) these temperatures equivalent. For example, in stellar interiors, where the plasma is very well-coupled, a single “temperature” defines each of the following: the velocity distribution, the ionization distribution, the spectrum, and the level populations. In the ISM each of these can be characterized by a different “temperature!”

Brightness Temperature

$T_B =$ the temperature of a blackbody that reproduces a given flux density at a specific frequency, such that

$B_\nu(T_B) = \frac{2 h \nu^3}{c^2} \frac{1}{{\rm exp}(h \nu / kT_B) - 1}$

Note: units for $B_{\nu}$ are ${\rm erg~cm^{-2}~s^{-1}~Hz^{-1}~ster^{-1}}$ .

This is a fundamental concept in radio astronomy. Note that the above definition assumes that the index of refraction in the medium is exactly 1.

Effective Temperature

$T_{\rm eff}$ (also called $T_{\rm rad}$ , the radiation temperature) is defined by

$\int_\nu B_\nu d\nu = \sigma T_{{\rm eff}}^4$ ,

which is the integrated intensity of a blackbody of temperature $T_{\rm eff}$ . $\sigma = (2 \pi^5 k^4)/(15 c^2 h^3)=5.669 \times 10^{-5} {\rm erg~cm^{-2}~s^{-1}~K^{-4}}$ is the Stefan-Boltzmann constant.

Color Temperature

$T_c$ is defined by the slope (in log-log space) of an SED. Thus $T_c$ is the temperature of a blackbody that has the same ratio of fluxes at two wavelengths as a given measurement. Note that $T_c = T_b = T_{\rm eff}$ for a perfect blackbody.

Kinetic Temperature

$T_k$ is the temperature that a particle of gas would have if its Maxwell-Boltzmann velocity distribution reproduced the width of a given line profile. It characterizes the random velocity of particles. For a purely thermal gas, the line profile is given by

$I(\nu) = I_0~e^{\frac{-(\nu-\nu_{jk})^2}{2\sigma^2}}$ ,

where $\sigma_{\nu}=\frac{\nu_{jk}}{c}\sqrt{\frac{kT_k}{\mu}}$ in frequency units, or

$\sigma_v=\sqrt{\frac{kT_k}{\mu}}$ in velocity units.

In the “hot” ISM $T_k$ is characteristic, but when $\Delta v_{\rm non-thermal} > \Delta v_{\rm thermal}$ (where $\Delta v$ are the Doppler full widths at half-maxima [FWHM]) then $T_k$ does not represent the random velocity distribution. Examples include regions dominated by turbulence.

$T_k$ can be different for neutrals, ions, and electrons because each can have a different Maxwellian distribution. For electrons, $T_k = T_e$ , the electron temperature.

Ionization Temperature

$T_I$ is the temperature which, when plugged into the Saha equation, gives the observed ratio of ionization states.

Excitation Temperature

$T_{\rm ex}$ is the temperature which, when plugged into the Boltzmann distribution, gives the observed ratio of two energy states. Thus it is defined by

$\frac{n_k}{n_j}=\frac{g_k}{g_j}~e^{-h\nu_{jk}/kT_{\rm ex}}$ .

Note that in stellar interiors, $T_k = T_I = T_{\rm ex} = T_c$ . In this room, $T_k = T_I = T_{\rm ex} \sim 300K$ , but $T_c \sim 6000K$ .

Spin Temperature

$T_s$ is a special case of $T_{\rm ex}$ for spin-flip transitions. We’ll return to this when we discuss the important 21-cm line of neutral hydrogen.

Bolometric temperature

$T_{\rm bol}$ is the temperature of a blackbody having the same mean frequency as the observed continuum spectrum. For a blackbody, $T_{\rm bol} = T_{\rm eff}$ . This is a useful quantity for young stellar objects (YSOs), which are often heavily obscured in the optical and have infrared excesses due to the presence of a circumstellar disk.

Antenna temperature

$T_A$ is a directly measured quantity (commonly used in radio astronomy) that incorporates radiative transfer and possible losses between the source emitting the radiation and the detector. In the simplest case,

$T_A = \eta T_B( 1 - e^{-\tau})$ ,

where $\eta$ is the telescope efficiency (a numerical factor from 0 to 1) and $\tau$ is the optical depth.

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equilibria, thermal (blackbody) emission, thermodynamics

CHAPTER: Important Properties of Local Thermodynamic Equilibrium

In Book Chapter on March 7, 2013 at 3:25 am

(updated for 2013)

For actual local thermodynamic equilbrium (not ETE), the following are important to keep in mind:

Detailed balance: transition rate from j to k = rate from k to j (i.e. no net change in particle distribution)
LTE is equivalent to ETE when $b_j = 1$ or $\frac{b_j}{b_k} = 1$
LTE is only an approximation, good under specific conditions.
Radiation intensity produced is not blackbody illumination as you’d want for true thermodynamic equilibrium.
Radiation is usually much weaker than the Planck function, which means not all levels are populated.
LTE assumption does not mean the Saha equation is applicable since radiative processes (not collisions) dominate in many ISM cases where LTE is applicable.

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equilibria, non-thermal processes, thermodynamics

CHAPTER: The Saha Equation

In Book Chapter on March 5, 2013 at 3:21 am

(updated for 2013)

How do we deal with the distribution over different states of ionization r? In thermodynamic equilibrium, the Saha equation gives:

$\frac{ n^\star(X^{(r+1)}) n_e } { n^\star (X^{(r)}) } = \frac{ f_{r+1} f_e}{f_r}$ ,

where $f_r$ and $f_{r+1}$ are the partition functions as discussed in the previous section. The partition function for electrons is given by

$f_e = 2\big( \frac{2 \pi m_e k T} {h^2} \big) ^{3/2} = 4.829 \times 10^{15} (\frac{T}{K})^{3/2}$

For a derivation of this, see pages 103-104 of this handout from Bowers and Deeming.

If $f_r$ and $f_{r+1}$ are approximated by the first terms in their sums (i.e. if the ground state dominates their level populations), then

$\frac{ n^\star ( X^{ (r+1) } ) n_e } {n^\star ( X^{ (r) } ) } = 2 \big(\frac{ g_{r+1,1} }{g_{ r,1}}\big) \big( \frac{ 2 \pi m_e k T} {h^2} \big)^{3/2} e^{-\Phi_r / kT}$ ,

where $\Phi_r=E_{r+1,1}-E_{r,1}$ is the energy required to ionize $X^{(r)}$ from the ground (j = 1) level. Ultimately, this is just a function of $n_e$ and $T$ . This assumes that the only relevant ionization process is via thermal collision (i.e. shocks, strong ionizing sources, etc. are ignored).

▶ Comment

emission/absorption of spectral lines, equilibria, non-thermal processes

CHAPTER: Spitzer Notation

In Book Chapter on March 5, 2013 at 3:19 am

(updated for 2013)

We will use the notation from Spitzer (1978). See also Draine, Ch. 3. We represent the density of a state j as

$n_j(X^{(r)})$ , where

n: particle density
j: quantum state
X: element
(r): ionization state
For example, $HI = H^{(0)}$

In his book, Spitzer defines something called “Equivalent Thermodynamic Equilibrium” or “ETE”. In ETE, $n_j^*$ gives the “equivalent” density in state j. The true (observed) value is $n_j$ . He then defines the ratio of the true density to the ETE density to be

$b_j = n_j / n_j^*$ .

This quantity approaches 1 when collisions dominate over ionization and recombination. For LTE, $b_j = 1$ for all levels. The level population is then given by the Boltzmann equation:

$\frac{n_j^\star(X^{(r)})}{n_k^\star(X^{(r)})} = (\frac{g_{rj}}{g_{rk}})~e^{ -(E_{rj} - E_{rk}) / kT }$ ,

where $E_{rj}$ and $g_{rj}$ are the energy and statistical weight (degeneracy) of level j, ionization state r. The exponential term is called the “Boltzmann factor”‘ and determines the relative probability for a state.

The term “Maxwellian” describes the velocity distribution of a 3-D gas. “Maxwell-Boltzmann” is a special case of the Boltzmann distribution for velocities.

Using our definition of b and dropping the “r” designation,

$\frac{n_k}{n_j} = \frac{b_k}{b_j} (\frac{g_k}{g_j})~e^{-h \nu_{jk} / kT }$

Where $\nu_{jk}$ is the frequency of the radiative transition from k to j. We will use the convention that $E_k > E_j$ , such that $E_{jk}=h\nu_{jk} > 0$ .

To find the fraction of atoms of species $X^{(r)}$ excited to level j, define:

$\sum_k n_k^\star (X^{(r)}) = n^\star(X^{(r)})$

as the particle density of $X^{(r)}$ in all states. Then

$\frac{ n_j^* (X^{(r)}) } { n^* (X^{(r)})} = \frac{ g_{rj} e^{-E_{rj} / kT} } {\sum_k g_{rk} e^{ -E_{rk} / kT} }$

Define $f_r$ , the “partition function” for species $X^{(r)}$ , to be the denominator of the RHS of the above equation. Then we can write, more simply:

$\frac{n_j^\star}{n^\star} = \frac{g_{rj}}{f_r} e^{-E_{rj}/kT}$

to be the fraction of particles that are in state j. By computing this for all j we now know the distribution of level populations for ETE.

▶ Comment

emission/absorption of spectral lines, equilibria, photons and radiation, thermodynamics

CHAPTER: Thermodynamic Equilibrium

In Book Chapter on February 28, 2013 at 3:13 am

(updated for 2013)

Collisions and radiation generally compete to establish the relative populations of different energy states. Randomized collisional processes push the distribution of energy states to the Boltzmann distribution, $n_j \propto e^{-E_j / kT}$ . When collisions dominate over competing processes and establish the Boltzmann distribution, we say the ISM is in Thermodynamic Equilibrium.

Often this only holds locally, hence the term Local Thermodynamic Equilibrium or LTE. For example, the fact that we can observe stars implies that energy (via photons) is escaping the system. While this cannot be considered a state of global thermodynamic equilibrium, localized regions in stellar interiors are in near-equilibrium with their surroundings.

But the ISM is not like stars. In stars, most emission, absorption, scattering, and collision processes occur on timescales very short compared with dynamical or evolutionary timescales. Due to the low density of the ISM, interactions are much more rare. This makes it difficult to establish equilibrium. Furthermore, many additional processes disrupt equilibrium (such as energy input from hot stars, cosmic rays, X-ray background, shocks).

As a consequence, in the ISM the level populations in atoms and molecules are not always in their equilibrium distribution. Because of the low density, most photons are created from (rare) collisional processes (except in locations like HII regions where ionization and recombination become dominant).

▶ Comment

atomic and molecular gas, dust, emission/absorption of spectral lines, photons and radiation, SED modeling

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:

Very rarely is a system actually in a true equilibrium state.
Except in HII regions, transitions in the ISM are usually not electronic.
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.
Transitions can be induced by photons, cosmic rays, collisions with atoms and molecules, and interactions with free electrons.
Levels can refer to electronic, rotational, vibrational, spin, and magnetic states.
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.

▶ Comment

magnetic fields, probes of velocity

CHAPTER: Relevant Velocities in the ISM

In Book Chapter on February 28, 2013 at 3:06 am

(updated for 2013)

Note: it’s handy to remember that 1 km/s ~ 1 pc / Myr.

Galactic rotation: 18 km/s/kpc (e.g. 180 km/s at 10 kpc)
Isothermal sound speed:
- For H, this speed is 0.3, 1, and 3 km/s at 10 K, 100 K, and 1000 K, respectively.
Alfvén speed: The speed at which magnetic fluctuations propagate. Alfvén waves are transverse waves along the direction of the magnetic field.
- Note that $v_A = {\rm const}$ if $B \propto \rho^{1/2}$ , which is observed to be true over a large portion of the ISM.
- Interstellar B-fields can be measured using the Zeeman effect. Observed values range from in the diffuse ISM to in dense clouds. For specific conditions:
  - $B = 1~\mu{\rm G}, n = 1 ~{\rm cm}^{-3} \Rightarrow v_A = 2~{\rm km~s}^{-1}$
  - $B = 30~\mu {\rm G}, n = 10^4~{\rm cm}^{-3} \Rightarrow v_A = 0.4~{\rm km~s}^{-1}$
  - $B = 1~{\rm mG}, n = 10^7 {\rm cm}^{-3} \Rightarrow v_A = 0.5~{\rm km~s}^{-1}$
- Compare to the isothermal sound speed, which is 0.3 km/s in dense gas at 20 K.
  - $c_s \approx v_A$ in dense gas
  - $c_s < v_A$ in diffuse gas
Observed velocity dispersion in molecular gas is typically about 1 km/s, and is thus supersonic. This is a signature of the presence of turbulence. (see the summary of Larson’s seminal 1981 paper)

▶ Comment

cosmic rays, photons and radiation

CHAPTER: Energy Density Comparison

In Book Chapter on February 26, 2013 at 3:04 am

(updated for 2013)

See Draine table 1.5. The primary sources of energy present in the ISM are:

The CMB ( $T_{\rm CMB}=2.725~{\rm K}$
Thermal IR from dust
Starlight ( $h\nu < 13.6 {\rm eV}$
Thermal kinetic energy (3/2 nkT)
Turbulent kinetic energy ( $1/2 \rho \sigma_v^2$ )
Magnetic fields ( $B^2 / 8 \pi$ )
Cosmic rays

All of these terms have energy densities within an order of magnitude of $1 ~{\rm eV ~ cm}^{-3}$ . With the exception of the CMB, this is not a coincidence: because of the dynamic nature of the ISM, these processes are coupled together and thus exchange energy with one another.

▶ Comment

atomic and molecular gas, cosmic rays, emission/absorption of spectral lines, ISM evidence/probes, ISM phases, photons and radiation, probes of composition, probes of density, probes of magnetic fields, probes of temperature, probes of velocity, radiative transfer, SED modeling, spectroscopy

CHAPTER: Measuring States in the ISM

In Book Chapter on February 26, 2013 at 3:00 am

(updated for 2013)

There are two primary observational diagnostics of the thermal, chemical, and ionization states in the ISM:

Spectral Energy Distribution (SED; broadband low-resolution)
Spectrum (narrowband, high-resolution)

SEDs

Very generally, if a source’s SED is blackbody-like, one can fit a Planck function to the SED and derive the temperature and column density (if one can assume LTE). If an SED is not blackbody-like, the emission is the sum of various processes, including:

thermal emission (e.g. dust, CMB)
synchrotron emission (power law spectrum)
free-free emission (thermal for a thermal electron distribution)

Spectra

Quantum mechanics combined with chemistry can predict line strengths. Ratios of lines can be used to model “excitation”, i.e. what physical conditions (density, temperature, radiation field, ionization fraction, etc.) lead to the observed distribution of line strengths. Excitation is controlled by

collisions between particles (LTE often assumed, but not always true)
photons from the interstellar radiation field, nearby stars, shocks, CMB, chemistry, cosmic rays
recombination/ionization/dissociation

Which of these processes matter where? In class (2011), we drew the following schematic.

A schematic of several structures in the ISM

Key

A: Dense molecular cloud with stars forming within

$T=10-50~{\rm K};~n>10^3~{\rm cm}^{-3}$ (measured, e.g., from line ratios)
gas is mostly molecular (low T, high n, self-shielding from UV photons, few shocks)
not much photoionization due to high extinction (but could be complicated ionization structure due to patchy extinction)
cosmic rays can penetrate, leading to fractional ionization: $X_I=n_i/(n_H+n_i) \approx n_i/n_H \propto n_H^{-1/2}$ , where $n_i$ is the ion density (see Draine 16.5 for details). Measured values for $X_e$ (the electron-to-neutral ratio, which is presumed equal to the ionization fraction) are about $X_e \sim 10^{-6}~{\rm to}~10^{-7}$ .
possible shocks due to impinging HII region – could raise T, n, ionization, and change chemistry globally
shocks due to embedded young stars w/ outflows and winds -> local changes in T, n, ionization, chemistry
time evolution? feedback from stars formed within?

B: Cluster of OB stars (an HII region ionized by their integrated radiation)

7000 < T < 10,000 K (from line ratios)
gas primarily ionized due to photons beyond Lyman limit (E > 13.6 eV) produced by O stars
elements other than H have different ionization energy, so will ionize more or less easily
HII regions are often clumpy; this is observed as a deficit in the average value of $n_e$ from continuum radiation over the entire region as compared to the value of n_e derived from line ratios. In other words, certain regions are denser (in ionized gas) than others.
The above introduces the idea of a filling factor, defined as the ratio of filled volume to total volume (in this case the filled volume is that of ionized gas)
dust is present in HII regions (as evidenced by observations of scattered light), though the smaller grains may be destroyed
significant radio emission: free-free (bremsstrahlung), synchrotron, and recombination line (e.g. H76a)
chemistry is highly dependent on n, T, flux, and time

C: Supernova remnant

gas can be ionized in shocks by collisions (high velocities required to produce high energy collisions, high T)
e.g. if v > 1000 km/s, T > 10⁶ K
atom-electron collisions will ionize H, He; produce x-rays; produce highly ionized heavy elements
gas can also be excited (e.g. vibrational H₂ emission) and dissociated by shocks

D: General diffuse ISM

UV radiation from the interstellar radiation field produces ionization
ne best measured from pulsar dispersion measure (DM), an observable. ${\rm DM} \propto \int n_e dl$
role of magnetic fields depends critically on X_I, n (B-fields do not directly affect neutrals, though their effects can be felt through ion-neutral collisions)

▶ Comment

chemistry, emission/absorption of spectral lines

CHAPTER: Chemistry

In Book Chapter on February 13, 2013 at 10:04 pm

See Draine Table 1.4 for elemental abundances for the proto-solar environment. H:He:C = $1:0.1:3 \times 10^{-4}$ by number. $1:0.4:3.5 \times 10^{-3}$ by mass However, these ratios vary by position in the galaxy, especially for heavier elements (which depend on stellar processing). For example, the abundance of heavy elements (Z > Carbon) is twice as low at the sun’s position than in the Galactic center. Even though metals account for 1% of the mass, they dominate most of the important chemistry, ionization, and heating/cooling processes. They are essential for star formation, as they allow molecular clouds to cool and collapse. Generally, it is easier (i.e. requires less energy) to dissociate a molecule than to ionize something. The lower the electronic state you are trying to ionize, the more energy is needed. The Lyman Limit is the minimum photon energy needed to ionize Hydrogen from the ground state (13.6 eV, 912 Angstrom).

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chemistry, emission/absorption of spectral lines, spectroscopy

CHAPTER: Hydrogen Slang

In Book Chapter on February 12, 2013 at 10:02 pm

Lyman limit: the minimum energy needed to remove an electron from a Hydrogen atom. A “Lyman limit photon” is a photon with at least this energy.

$E = 13.6 {\rm eV} = 1~ {\rm Rydberg} = hcR_{\rm H}$ ,

where $R_{\rm H}=1.097 \times 10^{7} {\rm m}^{-1}$ is the Rydberg constant, which has units of $1/\lambda$ . This energy corresponds to the Lyman limit wavelength as follows:

$E = h\nu = hc/\lambda \Rightarrow \lambda=912 \AA$ .

Lyman series: transitions to and from the n=1 energy level of the Bohr atom. The first line in this series was discovered in 1906 using UV studies of electrically excited hydrogen gas.

Balmer series: transitions to and from the n=2 energy level. Discovered in 1885; since these are optical transitions, they were more easily observed than the UV Lyman series transitions.

There are also other named series corresponding to higher n. Examples include Paschen (n=3), Brackett (n=4), and Pfund (n=5). The wavelength of a given transition can be computed via the Rydberg equation

$\frac{1}{\lambda}=R_{\rm H} \big(\frac{1}{n_f^2}-\frac{1}{n_i^2}\big)$ .

Note that the Lyman (or Balmer, Paschen, etc.) limit can be computed by inserting $n_i=\infty$ .

Lyman continuum corresponds to the region of the spectrum near the Lyman limit, where the spacing between energy levels becomes comparable to spectral line widths and so individual lines are no longer distinguishable.

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equilibria, excitation and ionization, ISM phases

CHAPTER: Stromgren Sphere: An example “chalkboard derivation”

In Book Chapter on February 12, 2013 at 8:31 pm

(updated for 2013)

The Stromgren sphere is a simplified analysis of the size of HII regions. Massive O and B stars emit many high-energy photons, which will ionize their surroundings and create HII regions. We assume that such a star is embedded in a uniform medium of neutral hydrogen. A sphere of radius r around this star will become ionized; r is called the “Stromgren radius”. The volume of the ionized region will be such that the rate at which ionized hydrogen recombines equals the rate at which the star emits ionizing photons (i.e. all of the ionizing photons are “used up” re-ionizing hydrogen as it recombines)

The recombination rate density is $\alpha n^2$ , where $\alpha$ is the recombination coefficient (in $\mathrm{cm}^3~\mathrm{s}^{-1})$ and $n=n_e=n_\mathrm{H}$ is the number density (assuming fully ionized gas and only hydrogen, the electron and proton densities are equal). The total rate of ionizing photons (in photons per second) in the volume of the sphere is $N^*$ . Setting the rates of ionization and recombination equal to one another, we get

$\frac43 \pi r^3 \alpha n^2 = N^*$ , and solving for r,

$r = ( \frac {3N^*} {4\pi\alpha n^2})^{\frac13}$

Typical values for the above variables are $N^* \sim 10^{49}~\mathrm{photons~s}^{-1}$ , $\alpha \sim 3\times 10^{-13}\; \mathrm{cm}^3 \; \mathrm s^{-1}$ and $n \sim 10\; \mathrm {cm}^{-3}$ , implying Stromgren radii of 10 to 100 pc. See the journal club (2013) article for discussion of Stromgren’s seminal 1939 paper.

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instabilities, shock physics, turbulence

CHAPTER: The Sound Speed

In Book Chapter on February 7, 2013 at 10:00 pm

(updated for 2013)

The speed of sound is the speed at which pressure disturbances travel in a medium. It is defined as

$c_s \equiv \frac{\partial P}{\partial \rho}$ ,

where $P$ and $\rho$ are pressure and mass density, respectively. For a polytropic gas, i.e. one defined by the equation of state $P \propto \rho^\gamma$ , this becomes $c_s=\sqrt{\gamma P/\rho}$ . $\gamma$ is the adiabatic index (ratio of specific heats), and $\gamma=5/3$ describes a monatomic gas.

For an isothermal gas where the ideal gas equation of state $P=\rho k_B T / (\mu m_{\rm H})$ holds, $c_s=\sqrt{k_B T/\mu}$ . Here, $\mu$ is the mean molecular weight (a factor that accounts for the chemical composition of the gas), and $m_{\rm H}$ is the hydrogen atomic mass. Note that for pure molecular hydrogen $\mu=2$ . For molecular gas with ~10% He by mass and trace metals, $\mu \approx 2.7$ is often used.

A gas can be approximated to be isothermal if the sound wave period is much higher than the (radiative) cooling time of the gas, as any increase in temperature due to compression by the wave will be immediately followed by radiative cooling to the original equilibrium temperature well before the next compression occurs. Many astrophysical situations in the ISM are close to being isothermal, thus the isothermal sound speed is often used. For example, in conditions where temperature and density are independent such as H II regions (where the gas temperature is set by the ionizing star’s spectrum), the gas is very close to isothermal.

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cosmic rays, equilibria, magnetic fields, photons and radiation

CHAPTER: Energy Density Comparison

In Book Chapter on February 6, 2013 at 10:06 pm

See Draine table 1.5 The main kinds of energy present in the ISM are:

The CMB
Thermal IR from dust
Starlight
Thermal kinetic energy (3/2 nKT)
Turbulent kinetic energy ( $1/2 \rho \sigma_v^2$ )
B fields ( $B^2 / 8 \pi$ )
Cosmic rays

All of these terms have energy densities within an order of magnitude of $1 ~{\rm eV ~ cm}^{-3}$ . With the exception of the CMB, this is not a coincidence. Because of the dynamic nature of the ISM, these processes are coupled together and thus exchange energy with one another.

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atomic and molecular gas, Galactic structure evidence, ISM phases

CHAPTER: Topology of the ISM

In Book Chapter on February 6, 2013 at 9:57 pm

(updated for 2013)

A grab-bag of properties of the Milky Way

HII scale height: 1 kpc
CO scale height: 50-75 pc
HI scale height: 130-400 pc
Stellar scale height: 100 pc in spiral arm, 500 pc in disk
Stellar mass: $5 \times 10^{10} M_\odot$
Dark matter mass: $5 \times 10^{10} M_\odot$
HI mass: $2.9 \times 10^9 M_\odot$
H₂ mass (inferred from CO): $0.84 \times 10^9 M_\odot$
HII mass: $1.12 \times 10^9~M_\odot$
-> total gas mass $= 6.7 \times 10^9~M_\odot$ (including He).
Total MW mass within 15 kpc: $10^{11} M_\odot$ (using the Galaxy’s rotation curve). About 50% dark matter.

So the ISM is a relatively small constituent of the Galaxy (by mass).

The Milky Way is a very thin disk (think a CD with a ping-pong ball in the middle) In class (2011), we drew the following schematic of the “topology” of phases in the ISM.

A schematic of several structures in the ISM

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atomic and molecular gas, cosmic rays, dust, ISM phases, magnetic fields, photons and radiation

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

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

(updated for 2013)

Gas
Dust
Cosmic Rays*
Photons**
B-Field
Gravitational Field
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:

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.
HII gas: Ionized mostly by O and early B stars. Called an “HII region” when confined by a molecular cloud, otherwise called “diffuse HII”.
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.
Cool HI (Cold Neutral Medium, or “CNM”). $T \sim 100~{\rm K}, n \sim 30 ~{\rm cm}^{-3}$ . Fills ~1% of the volume.
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 $10~{\rm to}~50~{\rm cm}^{-3}$ .
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).
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.

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atomic and molecular gas, cosmic rays, dust, magnetic fields

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 $\mu$ m) 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.

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extinction, ISM evidence/probes, spectroscopy

CHAPTER: How do we know there is an ISM?

In Book Chapter on January 29, 2013 at 8:35 pm

(updated for 2013)

Early astronomers pointed to 3 lines of evidence for the ISM:

Extinction. The ISM obscures the light from background stars. In 1919, Barnard (JC 2011, 2013) called attention to these “dark markings” on the sky, and put forward the (correct) hypothesis that these were the silhouettes of dark clouds. A good rule of thumb for the amount of extinction present is 1 magnitude of extinction per kpc (for typical, mostly unobscured lines-of-sight).
Reddening. Even when the ISM doesn’t completely block background starlight, it scatters it. Shorter-wavelength light is preferentially scattered, so stars behind obscuring material appear redder than normal. If a star’s true color is known, its observed color can be used to infer the column density of the ISM between us and the star. Robert Trumpler first used measurements of the apparent “cuspiness” and the brighnesses of star clusters in 1930 to argue for the presence of this effect. Reddening of stars of “known” color is the basis of NICER and related techniques used to map extinction today.
Stationary Lines. Spectral observations of binary stars show doppler-shifted lines corresponding to the radial velocity of each star. In addition, some of these spectra exhibit stationary (i.e. not doppler-shifted) absorption lines due to stationary material between us and the binary system. Johannes Hartmann first noticed this in 1904 when investigating the spectrum of $\delta$ Orionis: “The calcium line at $\lambda 3934$ [angstroms] exhibits a very peculiar behavior. It is distinguished from all the other lines in this spectrum, first by the fact that it always appears extraordinarily week, but almost perfictly sharp… Closer study on this point now led me to the quite surprising result that the calcium line… does not share in the periodic displacements of the lines caused by the orbital motion of the star”

Helpful References: Good discussion of the history of extinction and reddening, from Michael Richmond.

▶ Comment

IGM evidence, ISM phases, probes of density

CHAPTER: Density of the Intergalactic Medium

In Book Chapter on January 14, 2013 at 8:50 pm

(updated for 2013)

From cosmology observations, we know the universe to be very nearly flat ( $\Omega = 1$ ). This implies that the mean density of the universe is $\rho = \rho_{\rm crit} = \frac{3 H_0^2}{8 \pi G} = 7 \times 10^{-30} ~{\rm g~ cm}^{-3} \Rightarrow n<4.3 \times 10^{-6}~{\rm cm}^{-3}$ .

This places an upper limit on the density of the Intergalactic Medium.

▶ Comment

atomic and molecular gas, ISM phases, probes of density

CHAPTER: Density of the Milky Way’s ISM

In Book Chapter on January 14, 2013 at 8:46 pm

(updated for 2013)

How do we know that $n \sim 1 ~{\rm cm}^{-3}$ in the ISM? From the rotation curve of the Milky Way (and some assumptions about the mass ratio of gas to gas+stars+dark matter), we can infer

$M_{\rm gas} = 6.7 \times 10^{9} M_\odot$

Maps of HI and CO reveal the extent of our galaxy to be

$D = 40$ kpc

$h = 140$ pc (scale height of HI)

This applies an approximate volume of

$V = \pi D^2 h / 4 = 5 \times 10^{66} ~{\rm cm}^{3}$

Which, yields a density of

$\rho = 2.5 \times 10^{-24} ~{\rm g cm}^{-3}$

▶ Comment

ISM phases

CHAPTER: A Sense of Scale

In Book Chapter on January 13, 2013 at 8:40 pm

(updated for 2013)

How dense (or not) is the ISM?

Dense cores: $n \sim 10^5 ~{\rm cm}^{-3}$
Typical ISM: $n \sim 1 ~{\rm cm}^{-3}$
This room: 1 mol / 22.4L $\sim 3 \times 10^{19}~ {\rm cm}^{-3}$
XVH (eXtremely High Vacuum) — best human-made vacuum: $n \sim 3 \times 10^{4}~ {\rm cm}^{-3}$
Density of stars in the Milky Way: $2.8~{\rm stars/pc}^3 \approx 0.125~M_\odot/{\rm pc}^3 = 8.5 \times 10^{-24} ~{\rm g / cm}^3 \sim 5~{\rm cm}^{-3}$

In other words, most of the ISM is at a density far below the densities and pressures we can reproduce in the lab. Thus, the details of most of the microphysics in the ISM are still poorly understood. We also see that the density of stars in the Galaxy is quite small – only a few times the average particle density of the ISM.

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Archive for the ‘Book Chapter’ Category

Brightness Temperature

Effective Temperature

Color Temperature

Kinetic Temperature

Ionization Temperature

Excitation Temperature

Spin Temperature

Bolometric temperature

Antenna temperature

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