# Harvard Astronomy 201b

## 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 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.

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:
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.

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.

## 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)

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.

## 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.  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).

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.

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!”

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

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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
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Goodman, A. A., Barranco, J. A., Wilner, D. J., & Heyer, M. H. 1998, ApJ, 504, 223
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## ARTICLE: The Physical State of Interstellar Hydrogen

In Journal Club 2013 on February 12, 2013 at 9:57 pm

The Physical State of Interstellar Hydrogen by Bengt Strömgren (1939)

Summary by Anjali Tripathi

Abstract

In 1939, Bengt Strömgren published an analytic formulation for the spatial extent of ionization around early type stars.  Motivated by new H-alpha observations of sharply bound “diffuse nebulosities,” Strömgren was able to characterize these ionized regions and their thin boundaries in terms of the ionizing star’s properties and abundances of interstellar gas.  Strömgren’s work on these regions, which have come to be eponymously known as Strömgren spheres, has found longstanding use in the study of HII regions, as it provides a simple analytic approach to recover the idealized properties of such systems.

Background: Atomic Physics in Astronomy & New Observations

Danish astronomer Bengt Strömgren (1908-87) was born into a family of astronomers and educated during a period of rapid development in our understanding of the atom and modern physics.  These developments were felt strongly in Copenhagen where Strömgren studied and worked for much of his life.  At the invitation of Otto Struve, then director of Yerkes Observatory, Strömgren visited the University of Chicago from 1936 to 1938, where he encountered luminaries from across astrophysics, including Chandrasekhar and Kuiper.  With Struve and Kuiper, Strömgren worked to understand how photoionization could explain observations of a shell of gas around an F star, part of the eclipsing binary $\epsilon$ Aurigae (Kuiper, Struve and Strömgren, 1937).  This work laid out the analytic framework for a bounded region of ionized gas around a star, which provided the theoretical foundation for Strömgren’s later work on HII regions.

The observational basis for Strömgren’s 1939 paper came from new spectroscopic measurements taken by Otto Struve.  Using the new 150-Foot Nebular Spectrograph (Struve et al, 1938) perched on a slope at McDonald Observatory, pictured below, Struve and collaborators were able to resolve sharply bound extended regions “enveloped in diffuse nebulosities” in the Balmer H-alpha emission line (Struve and Elvey, 1938).  This emission line results from recombination when electrons transition from the n = 3 to n = 2 energy level of hydrogen, after the gas was initially ionized by UV radiation from O and B stars.  Comparing these observations to those of the central parts of the Orion Nebula led the authors to estimate that the number density of hydrogen with electrons in the n=3 state is $N_3 = 3 \times 10^{-21} cm^{-3}$, assuming a uniform concentration of stars and neglecting self-absorption (Struve and Elvey, 1938).  From his earlier work on $\epsilon$ Aurigae, Strömgren had an analytic framework with which to understand these observations.

Instrument used to resolve HII Regions in H-alpha (Struve et al, 1938)

Putting it together – Strömgren’s analysis

To understand the new observations quantitatively, Strömgren worked out the size of these emission nebulae by finding the extent of the ionized gas around the central star.  As in his paper with Kuiper and Struve, Strömgren considered only neutral and ionized hydrogen, assumed charge neutrality, and used the Saha equation with additional terms:

${N'' N_e \over N'} = \underbrace{{(2 \pi m_e)^{3/2} \over h^3} {2q'' \over q'} (kT)^{3/2}e^{-I/kT}}_\text{Saha} \cdot \underbrace{\sqrt{T_{el} \over T}}_\text{Temperature correction} \cdot \underbrace{R^2 \over 4 s^2}_\text{Geometrical Dilution}\cdot \underbrace{e^{-\tau_u}}_\text{Absorption}\\ N': \text{Neutral hydrogen (HI) number density}\\ N'':\text{Ionized hydrogen (HII) number density}\\ N_e:\text{Electron number density, }N_e = N''\text{ by charge neutrality}\\ x: \text{Ionization fraction}, x = N''/(N'+N'')$

Here, the multiplicative factor of $\sqrt{T_{el} \over T}$ corrects for the difference between the stellar temperature($T$) and the electron temperature($T_{el}$) at a distance $s$ away from the star.  The dilution factor ${R^2 \over 4 s^2}$, where $R$ is the stellar radius and $s$ is the distance from the star, accounts for the decrease in stellar flux with increasing distance.  The factor of $e^{-\tau_u}$, where $\tau_u$ is the optical depth, accounts for the reduction in the ionizing radiation due to absorption.  Taken together, this equation encapsulates the physics of a star whose photons ionize surrounding gas.  This ionization rate is balanced by the rate of recombination of ions and electrons to reform neutral hydrogen.  As a result, close to the star where there is abundant energetic flux, the gas is fully ionized, but further from the star, the gas is primarily neutral.  Strömgren’s formulation allowed him to calculate the location of the transition from ionized to neutral gas and to find the striking result that the transition region between the two is incredibly sharp, as plotted below.

Plot of ionization fraction vs. distance for an HII Region (Values from Table 2 of Strömgren, 1939)

Strömgren found that the gas remains almost completely ionized until a critical distance $s_0$, where the ionization fraction sharply drops and the gas becomes neutral due to absorption.  This critical distance has become known as the Strömgren radius, considered to be the radius of an idealized, spherical HII region.  The distance over which the ionization fraction drops from 1 to 0 is small (~0.01 pc), corresponding to one mean free path of an ionizing photon, compared to the Strömgren radius(~100pc).  Thus Strömgren’s analytic work provided an explanation for sharply bound ionized regions with thin transition zones separating the ionized gas from the exterior neutral gas.

Strömgren also demonstrated how the critical distance depends on the total number density $N$, the stellar effective temperature $T$, and the stellar radius $R$:

$\log{s_0} = -6.17 + {1 \over 3} \log{ \left( {2q'' \over q'} \sqrt{T_{el} \over T} \right)} - {1 \over 3} \log{a_u} - {1 \over 3} \frac{5040K}{T} I + {1 \over 2} \log{T} + {2 \over 3} \log{R} - {2 \over 3} \log{N},$

where $a_u$ is the absorption coefficient for the ionizing radiation per hydrogen atom (here assumed to be frequency independent) and $s_0$ is given in parsecs.  From this relation, we can see that for a given stellar radius and a fixed number density, $s_0 \propto T^{1/2}$, so that hotter, earlier type stars have larger ionized regions.  Plugging in numbers, Strömgren found that for a total number density of $3~cm^{-3}$, a cluster of 10 O7 stars would have a critical radius of 100-150 parsecs, in agreement with estimates made by the Struve and Elvey observations.

To estimate the hydrogen number density from the H-alpha observations, Strömgren also considered the excitation of the n=3 energy levels of hydrogen.  Weighing the relative importance of various mechanisms for excitation – free electron capture, Lyman-line absorption, Balmer-line absorption, and collisions – Strömgren found that their effects on the number densities of the excited states and electron number densities were comparable.  As a result, he estimated from Struve’s and Elvey’s $N_3$ that the number density of hydrogen is 2-3 $cm^{-3}$.

Strömgren’s analysis of ionized regions around stars and neutral hydrogen in “normal regions” matched earlier theoretical work by Eddington into the nature of the ISM (Eddington, 1937).  “With great diffidence, having not yet rid myself of the tradition that ‘atoms are physics, but molecules are chemistry’,” Eddington wrote that “presumably a considerable part” of the ISM is molecular.  As a result, Strömgren outlined how his analysis for ionization regions could be modified to consider regions of molecular hydrogen dissociating, presciently leaving room for the later discovery of an abundance of molecular hydrogen.  Instead of the ionization of atomic hydrogen, Strömgren worked with the dissociation of molecular hydrogen in this analysis.   Given that the energy required to dissociate the bond of molecular hydrogen is less than that required to ionize atomic hydrogen, Stromgren’s analysis gives a model of a star surrounded by ionized atoms, which is surrounded by a sharp, thin transition region of atomic hydrogen, around which molecular hydrogen remains.

In addition to HI and HII, Strömgren also considered the ionization of other atoms and transitions.  For example, Strömgren noted that if the helium abundance was smaller than that of hydrogen, then most all of the helium will be ionized out to the boundary of the hydrogen ionization region.  From similar calculations and considering the observations of Struve and Elvey, Strömgren was able to provide an estimate of the abundance of OII, a ratio of $10^{-2}-10^{-3}$ oxygen atoms to each hydrogen atom.

Strömgren Spheres Today

Strömgren’s idealized formulation for ionized regions around early type stars was well received initially and  has continued to influence thinking about HII regions in the decades since.  The simplicity of Strömgren’s model and its assumptions, however, have been recognized and addressed over time.  Amongst these are concerns about the assumption of a uniformly dense medium around the star.  Optical and radio observations, however, have revealed that the surrounding nebula can have clumps and voids – far from being uniformly dense (Osterbrock and Flather, 1959).  To address this, calculations of the nebula’s density can include a ‘filling factor’ term.  Studies of the Orion Nebula (M42), pictured below, have provided examples of just such clumpiness.  M42 has also been used to study another related limitation of Strömgren’s model – the assumption of a central star surrounded by spherical symmetry.

Orion Nebula, infrared image from WISE. Credit: NASA/JPL/Caltech

Consideration of the geometry of Strömgren spheres has been augmented by blister models of the 1970s whereby a star ionizes surrounding gas but the star is at the surface or edge of a giant molecular cloud (GMC), rather than at the center of it.  As a result, ionized gas breaks out of the GMC, like a popping blister, which in turn can prompt “champagne flows” of ionized gas leaching into the surrounding medium.  In a review article of Strömgren’s work, Odell (1999) states that due to observational selection effects, many HII regions observed in the optical actually are more akin to blister regions, rather than Strömgren spheres, since Strömgren spheres formed at the center or heart of a GMC may be obscured so much that they are observable only at radio wavelengths.

In spite of its simplifying assumptions, Strömgren’s work remains relevant today.   Given its abundance, hydrogen dominates the physical processes of emission nebulae and, thus, Strömgren’s idealized model provides a good first approximation for the ionization structure, even though more species are involved than just atomic hydrogen.  Today we can enhance our understanding of these HII regions using computer codes, such as CLOUDY, to calculate the ionization states of various atoms and molecules.  We can also computationally model the  hydrodynamics  of shocks radiating outwards from the star and use spectral synthesis codes to produce mock spectra.  From these models and the accumulated wealth of observations over time, we have come to accept that dense clouds of molecular gas, dominated with molecular hydrogen, are the sites of star formation.  Young O and B-type stars form out of clumps in these clouds and their ionizing radiation will develop into an emission nebula with ionized atomic hydrogen, sharply bound from the surrounding neutral cloud.  As the stars age and the shocks race onwards, the HII regions will evolve.  What remains, however, is Strömgren’s work which provides a simple analytic basis for understanding the complex physics of HII regions.

Strömgren, “The Physical State of Interstellar Hydrogen”, ApJ (1939)

Kuiper et al, “The Interpretation of $\epsilon$  Aurigae”, ApJ (1937)

Struve et al, “The 150-Foot Nebular Spectrograph of the McDonald Observatory”, ApJ (1938)

Eddington, “Interstellar Matter”, The Observatory (1937)

Osterbrock and Flather, “Electron Densities in the Orion Nebula. II”, ApJ (1959)

O’Dell, “Strömgren Spheres”, ApJ (1999)

## 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; 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.