Harvard Astronomy 201b

Posts Tagged ‘thermodynamics’

ARTICLE: A Theory of the Interstellar Medium: Three Components Regulated by Supernova Explosions in an Inhomogeneous Substrate

In Journal Club 2013 on March 15, 2013 at 11:09 pm

Abstract (the paper’s, not ours)

Supernova explosions in a cloudy interstellar medium produce a three-component medium in which a large fraction of the volume is filled with hot, tenuous gas.  In the disk of the galaxy the evolution of supernova remnants is altered by evaporation of cool clouds embedded in the hot medium.  Radiative losses are enhanced by the resulting increase in density and by radiation from the conductive interfaces between clouds and hot gas.  Mass balance (cloud evaporation rate = dense shell formation rate) and energy balance (supernova shock input = radiation loss) determine the density and temperature of the hot medium with (n,T) = (10^{-2.5}, 10^{5.7}) being representative values.  Very small clouds will be rapidly evaporated or swept up.  The outer edges of “standard” clouds ionized by the diffuse UV and soft X-ray backgrounds provide the warm (~10^{4} K) ionized and neutral components.  A self-consistent model of the interstellar medium developed herein accounts for the observed pressure of interstellar clouds, the galactic soft X-ray background, the O VI absorption line observations, the ionization and heating of much of the interstellar medium, and the motions of the clouds.  In the halo of the galaxy, where the clouds are relatively unimportant, we estimate (n,T) = (10^{-3.3}, 10^{6.0}) below one pressure scale height.  Energy input from halo supernovae is probably adequate to drive a galactic wind.

The gist

The paper’s (McKee and Ostriker 1977) main idea is that supernova remnants (SNRs) play an important role in the regulation of the ISM.  Specifically, they argue that these explosions add enough energy that another phase is warranted: a Hot Ionized Medium (HIM)

A Bit About SNRs…

A basic supernova explosion consists of several phases.  Their characteristic energies are on the order of 10^{51} erg, and indeed this is a widely-used unit.  For a fairly well-characterized SNR, see Cas A which exploded in the late 1600s.

Nearby supernova remnant Cassiopeia A, in X-rays from NuSTAR.

Nearby supernova remnant Cassiopeia A, in X-rays from NuSTAR.

  1. Free expansion
    A supernova explosion begins by ejecting mass with a range of velocities, the rms of which is highly supersonic.  This means that a shock wave propagates into the ISM at nearly constant velocity during the beginning.  Eventually the density decreases and the shocked pressure overpowers the thermal pressure in the ejected material, creating a reverse shock propagating inwards.  This phase lasts for something on the order of several hundred years.  Much of the Cas A ejecta is in the free expansion phase, and the reverse shock is currently located at 60% of the outer shock radius.
  2. Sedov-Taylor phase
    The reverse shock eventually reaches the SNR center, the pressure of which is now extremely high compared to its surroundings.  This is called the “blast wave” portion, in which the shock propagates outwards and sweeps up material into the ISM.  The remnant’s time evolution now follows the Sedov-Taylor solution, which finds R_s \propto t^{2/5}.  This phase ends when the radiative losses (from hot gas interior to the shock front) become important.  We expect this phase to last about 10^3 years.
  3. Snowplow phase
    When the age of the SNR approaches the radiative cooling timescale, cooling causes thermal pressure behind the shock to drop, stalling it.  This phase features a shell of cool gas around a hot volume, the mass of which increases as it sweeps up the surrounding gas like a gasplow.  For typical SNRs, this phase ends at an age of about 10^6 yr, leading into the next phase:
  4. Fadeaway
    Eventually the shock speed approaches the sound speed in the gas, and turns into a sound wave.  The “fadeaway time” is on the order of 10^{6} years.

So why are they important?

To constitute an integral part of a model of the ISM, SNRs must occur fairly often and overlap.  In the Milky Way, observations indicate a supernova every 40 years.  Given the size of the disk, this yields a supernova rate of 10^{-13} pc^{-3} yr^{-1}.

Here we get some justification for an ISM that’s a bit more complicated than the then-standard two-phase model (proposed by Field, Goldsmith, and Habing (1969) consisting mostly of warm HI gas).  Taking into account the typical fadeaway time of a supernova, we can calculate that on average 1 other supernova will explode within a “fadeaway volume” within that original lifetime.  That volume is just the characteristic area swept out by the shock front as it approaches the sound speed in the last phase.  For a fadeaway time of 10^6 yr and a typical sound speed of the ISM, this volume is about 100 pc.  Thus in just a few million years, this warm neutral medium will be completely overrun by supernova remnants!  The resulting medium would consist of low-density hot gas and dense shells of cold gas.  McKee and Ostriker saw a better way…

The Three Phase Model

McKee and Ostriker present their model by following the evolution of a supernova remnant, eventually culminating in a consistent picture of the phases of the ISM. Their model consists of a hot ionized medium with cold dense clouds dispersed throughout. The cold dense clouds have surfaces that are heated by hot stars and supernova remnants, making up the warm ionized and neutral media, leaving the unheated interiors as the cold neutral medium. In this picture, supernova remnants are contained by the pressure of the hot ionized medium, and eventually merge with it. In the early phases of their expansion, supernova remnants evaporate the cold clouds, while in the late stages, the supernova remnant material cools by radiative losses and contributes to the mass of cold clouds.

McKee and Ostriker Three Phase Model
A schematic of the three phase model, showing how supernovae drive the evolution of the interstellar medium.

In the early phases of the supernova remnant, McKee and Ostriker focus on the effects of electron-electron thermal conduction. First, they cite arguments by Chevalier (1975) and Solinger, Rappaport, and Buff (1975) that conduction is efficient enough to make the supernova remnant’s interior almost isothermal. Second, they consider conduction between the supernova remnant and cold clouds that it engulfs. Radiative losses from the supernova remnant are negligible in this stage, so the clouds are evaporated and incorporated into the remnant. Considering this cloud evaporation, McKee and Ostriker modify the Sedov-Taylor solution for this stage of expansion, yielding two substages. In the first substage, the remnant has not swept up much mass from the hot ionized medium, so mass gain from evaporated clouds dominates. They show this mechanism actually modifies the Sedov-Taylor solution to a t^{3/5} dependance. In the second substage, the remnant has cooled somewhat, decreasing the cloud evaporation, making mass sweep-up the dominant effect. The classic t^{2/5} Sedov-Taylor solution is recovered.

The transition to the late stages occurs when the remnant has expanded and cooled enough that radiative cooling becomes important. Here, McKee and Ostriker pause to consider the properties of the remnant at this point (using numbers they calculate in later sections): the remnant has an age of 800 kyr, radius of 180 pc, density of 5 \times 10^{-3} cm^{-3}, and temperature of 400 000 K. Then, they consider effects that affect the remnant’s evolution at this stage:

  • When radiative cooling sets in, a cold, dense shell is formed by runaway cooling: in this regime, radiative losses increase as temperature decreases. This effect is important at a cooling radius where the cooling time equals the age of the remnant.
  • When the remnant’s radius is larger than the scale height of the galaxy, it could contribute matter and energy to the halo.
  • When the remnant’s pressure is comparable to the pressure of the hot ionized medium, the remnant has merged with the ISM.
  • If supernovae happen often enough, two supernova remnants could overlap.
  • After the cold shell has developed, when the remnant collides with a cold cloud, it will lose shell material to the cloud.

Frustratingly, they find that these effects become important at about the same remnant radius. However, they find that radiative cooling sets in slightly before the other effects, and continue to follow the remnant’s evolution.

The mean free path of the remnant’s cold shell against cold clouds is very short, making the last effect important once radiative cooling has set in. The shell condenses mass onto the cloud since the cloud is more dense, creating a hole in the shell. The density left behind in the remnant is insufficient to reform the shell around this hole. The radius at which supernova remnants are expected to overlap is about the same as the radius where the remnant is expected to collide with its first cloud after having formed a shell. Then, McKee and Ostriker state that little energy loss occurs when remnants overlap, and so the remnant must merge with the ISM here.

At this point, McKee and Ostriker consider equilibrium in the ISM as a whole to estimate the properties of the hot ionized medium in their model. First, they state that when remnants overlap, they must also be in pressure equilibrium with the hot ionized medium. Second, the remnants have added mass to the hot ionized medium by evaporating clouds and removed mass from the hot ionized medium by forming shells – but there must be a mass balance. This condition implies that the density of the hot ionized medium must be the same as the density of the interior of the remnants on overlap. Third, they state that the supernova injected energy that must be dissipated in order for equilibrium to hold. This energy is lost by radiative cooling, which is possible as long as cooling occurs before remnant overlap. Using supernovae energy and occurrence rate as well as cold cloud size, filling factor, and evaporation rate, they calculate the equilibrium properties of the hot ionized medium. They then continue to calculate “typical” (median) and “average” (mean) properties, using the argument that the hot ionized medium has some volume in equilibrium, and some volume in expanding remnants. They obtain a typical density of 3.5 \times 10^{-3} cm^{-3}, pressure of 5.0 \times 10^{-13} cm^{-2} dyn, and temperature of 460 000 K.

McKee and Ostriker also use their model to predict different properties in the galactic halo. There are fewer clouds, so a remnant off the plane would not gain as much mass from evaporating clouds. Since the remnant is not as dense, radiative cooling sets in later – and in fact, the remnant comes into pressure equilibrium in the halo before cooling sets in. Supernova thus heat the halo, which they predict would dissipate this energy by radiative cooling and a galactic wind.

Finally, McKee and Ostriker find the properties of the cold clouds in their model, starting from assuming a spectrum of cloud sizes. They use Hobbs’s (1974) observations that the number of clouds with certain column density falls with the column density squared, adding an upper mass limit from when the cloud exceeds the Jeans mass and gravitationally collapses. A lower mass limit is added from considering when a cloud would be optically thin to ionizing radiation. Then, they argue that the majority of the ISM’s mass lies in the cold clouds. Then using the mean density of the ISM and the production rate of ionizing radiation, they can find the number density of clouds and how ionized they are.

Parker Instability

The three-phase model gives little prominence to magnetic fields and giant molecular clouds. As a tangent from McKee and Ostriker’s model, the Parker model (Parker 1966) will be presented briefly to showcase the variety of considerations that can go into modelling the ISM.

The primary motivation for Parker’s model are observations (from Faraday rotation) that the magnetic field of the Galaxy is parallel to the Galactic plane. He also assumes that the intergalactic magnetic field is weak compared to the galactic magnetic field: that is, the galactic magnetic field is confined to the galaxy. Then, Parker suggests what is now known as the Parker instability: that instabilities in the magnetic field cause molecular cloud formation.

Parker’s argument relies on the virial theorem: in particular, that thermal pressure and magnetic pressure must be balanced by gravitational attraction. Put another way, field lines must be “weighed down” by the weight of gas they penetrate: if gravity is too weak, the magnetic fields will expand the gas it penetrates. Then, he rules out topologies where all field lines pass through the center of the galaxy and are weighed down only there: the magnetic field would rise rapidly towards the center, disagreeing with many observations. Thus, if the magnetic field is confined to the galaxy, it must be weighed down by gas throughout the disk.

He then considers a part of the disk, and assumes a uniform magnetic field, and shows that it is unstable to transverse waves in the magnetic field. If the magnetic field is perturbed to rise above the galactic plane, the gas it penetrates will slide down the field line towards the disk because of gravity. Then, the field line has less weight at the location of the perturbation, allowing magnetic pressure to grow the perturbation. Using examples of other magnetic field topologies, he argues that this instability is general as long as gravity is the force balancing magnetic pressure. By this instability, he finds that the end state of the gas is in pockets spaced on the order of the galaxy’s scale height. He suggests that this instability explains giant molecular cloud formation. The spacing between giant molecular clouds is of the right order of magnitude. Also, giant molecular clouds are too diffuse to have formed by gravitational collapse, whereas the Parker instability provides a plausible mode of formation.

In today’s perspective, it is thought that the Parker instability is indeed part of giant molecular cloud formation, but it is unclear how important it is. Kim, Ryu, Hong, Lee, and Franco (2004) collected three arguments against Parker instability being the sole cause:

  • The formation time predicted by the Parker instability is ~10 Myr. However, looking at giant molecular clouds as the product of turbulent flows gives very short lifetimes (Ballesteros-Paredes et al. 1999). Also, ~10 Myr post T Tauri stars are not found in giant molecular clouds, suggesting that they are young (Elmegreen 2000).
  • Adding a random component to the galactic magnetic field can stabilize the Parker instability (Parker & Jokipii 2000, Kim & Ryu 2001).
  • Simulations suggest that the density enhancement from Parker instability is too small to explain GMC formation (Kim et al. 1998, 2001).

Does it hold up to observations?

The paper offers several key observations justifying the model.  First, of course, is the observed supernova rate which argues that a warm intercloud medium would self-destruct in a few Myr.  Other model inputs include the energy per supernova, the mean density of the ISM, and the mean production rate of UV photons.

They also cite O VI absorption lines and soft X-ray emission as evidence of the three-phase model.  The observed oxygen line widths are a factor of 4 smaller than what would be expected if they originated in shocks or the Hot Ionized Medium, and they attribute this to the idea that the lines are generated in the conductive surfaces of clouds — a key finding of their model above.  If one observes soft X-ray emission across the sky, a hot component of T ~ 10^{6.3} K can be seen in data at 0.4-0.85 keV, which cannot be well explained just with SNRs of this temperature (due to their small filling factor).  This is interpreted as evidence for large-scale hot gas.

So can it actually predict anything?

Sure!  Most importantly, with just the above inputs — the supernova rate, the energy per supernova, and the cooling function — they are able to derive the mean pressure of the ISM (which they predict to be 3700 K cm^-3, very close to the observed thermal pressures).

Are there any weaknesses?

The most glaring omission of the three-phase model is that the existence of large amounts of warm HI gas, seen through 21cm emission, is not well explained; they underpredict the fraction of hydrogen in this phase by a factor of 15!  In addition, observed cold clouds are not well accounted for; they should disperse very quickly even at temperatures far below that of the ISM that they predict.

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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 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.

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.

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

      1. Coulomb force-dominated (r^{-1} potential): electron-ion, electron-electron, ion-ion
      2. Ion-neutral: induced dipole in neutral atom leads to r^{-4} potential; e.g. electron-neutral scattering
      3. 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.

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.

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.

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).

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).