Harvard Astronomy 201b

ARTICLE: A theory of the interstellar medium – Three components regulated by supernova explosions in an inhomogeneous substrate

In Journal Club, Journal Club 2011 on February 12, 2011 at 1:25 am

Read the paper by C.F. McKee & J.P. Ostriker (1977)

Summary by: Tanmoy Laskar

Abstract

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. 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.

Prevailing Paradigm – The FGH model

In 1969, Field, Goldsmith and Habing balanced heating (by cosmic rays) with cooling (by line emission) in the interstellar medium and deduced a phase diagram, plotting pressure versus density:
They find three phases, F (~10,000 K), G (~1000 K) and H (~100 K). Phase G is unstable (since a tiny compression or rarefaction leads even away from equilibrium), and so the ISM must be composed of two phases – cold and dense clouds (phase H), embedded in and in pressure equilibrium with a warm diffuse medium (phase F).

Key Idea of McKee and Ostriker

It turns out that energy injection by supernova explosions is an important contributor to the heating of the ISM, and produces a whole new phase! McKee and Ostriker performed a simple calculation using the rates of Supernova explosions in our Galaxy, and deduced that the all-permeating diffuse phase F above would “self-destruct”. They argued that the ISM really consists of three phases – the cold clouds of Lyman Spitzer and others,
surrounded by a warm (ionized) medium, immersed in a hot (~100,000 K) inter-cloud medium essentially formed from overlapping supernova blast waves. Thus they added a new phase to our view of the ISM.

Although many specific details in this model are now known to be inaccurate, the basic premise that the energy injected by supernova explosions is able to sustain a unique phase of the ISM (the HIM), has stood the test of time. This, is the story of their intellectual discovery.

The paper is quite technical, and it would help as we attempt to understand their arguments, if we chart out a path for ourselves. Here is the route we are going to follow
(and here’s a PDF version).

Flowchart to guide the reader through McKee and Ostriker (1997)

Stages of a Supernova Explosion

Let us begin by reviewing the stages of a supernova explosion, as relevant from the point of view of its impact on the ISM. These are:

  1. Ejecta-Dominated
    The supernova ejecta are in free expansion. The internal pressure is enormous compared to the ISM pressure, and stuff just goes ballistic. The ejecta rush out at speeds much larger than the sound speed of the medium, and a shock wave (the blast wave, or forward shock) propagates into the medium just in front of the ejecta. As the ejecta hits the ISM, it slows down, and stuff behind it bumps into it, setting up a reverse shock propagating back into the ejecta. This phase lasts up to about 200 years and ends when the blast wave shock sweeps up material of the order of the ejecta mass M_{\rm ej}. This phase is not relevant for McKee & Ostriker (1977).
  2. Sedov-Taylor (aka ‘Blastwave phase’)
    Once the reverse shock reaches the center of the ejecta, the remnant undergoes self-similar evolution. This solution was first proposed by Sedov and Taylor separately in the 1950s, and is described by the energy of the explosion, E_{\rm ej}, the ejecta mass, M_{\rm ej} and the time since the beginning of this phase, t. The blast wave shock is still propagating into the ISM, slowing down as it sweeps up material. Most of the ejecta is optically thick, and is not radiating efficiently.
  3. Snowplow, which can further be divided into
    1. Pressure-Driven Snowplow
    2. Momentum-Driven Snowplow
    3. The Sedov-Taylor (ST) phase ends when the cooling time of the ejecta becomes comparable to the age of the remnant (~5\times 10^4 yr), at which point it cools catastrophically. Material at the edge of the remnant cools first, and the interior is over-pressured, resulting in accumulation of matter in a thin shell behind the blast wave shock, which continues plowing into the ISM. Both the ST and snowplow phases are relevant for this paper.

The full details of supernova stages are discussed elsewhere.

Overlap Rates

Consider a peppering of supernova remnants throughout the ISM. Let each supernova be characterised by a radius, R, which depends simply on time, t as R = t^{\alpha}.

We now define the size distribution of the supernovae: the number of remnants with radius, R and volume, V(R) that overlap a given point per unit radius of the remnant per unit volume of the ISM is given by N(R) = \frac{{\rm d}^2N}{{\rm d}V {\rm d}R} V(R) .

Let Q be the average number of supernova remnants (of all radii) that overlap a given point. Q tells us how tightly packed supernova remnants are, and is given by
Q  = \int_0^{\infty} N(R) {\rm d}R  = \int_0^{\infty} \frac{{\rm d}^2N}{{\rm d}V {\rm d}R} V(R) {\rm d}R
= \int_0^{\infty} \frac{{\rm d}^2N}{{\rm d}V {\rm d}t} \frac{{\rm d}t}{{\rm d}R}  \frac{4\pi}{3}R^3 {\rm d}R  = \frac{4\pi}{3} \int_0^{\infty} S \frac{{\rm d}t}{{\rm d}R} R^3 {\rm d}R ,
where we have written
S = \frac{{\rm d}^2 N}{{\rm d}V {\rm d}t} , the local supernova rate per unit volume.

Now, if R \propto t^{\eta} (for definiteness, take R = At^{\eta}), then
\frac{{\rm d} \ln{R}}{{\rm d} \ln{t}} = \eta  \implies \frac{{\rm d} R}{{\rm d} t} = \frac{\eta R}{t}  \implies \frac{{\rm d} t}{{\rm d} R} = \frac{1}{\eta}\frac{t}{R}  = \frac{1}{\eta R} \left(\frac{R}{A}\right)^{\frac{1}{\eta}} ,
so that Q = \frac{4\pi}{3} \int_0^{\infty} \frac{S}{\eta R} \left(\frac{R}{A}\right)^{\frac{1}{\eta}} R^3 {\rm d}R  = \frac{4\pi}{3} \frac{S}{\eta} A^{-\frac{1}{\eta}}  \int_0^{\infty} R^{\frac{2\eta + 1}{\eta}} {\rm d}R
In the regime where \eta \in (-\frac{1}{3},0), this is integrable and gives
Q = \frac{4\pi}{3} \frac{S}{\eta} A^{-\frac{1}{\eta}} \eta \frac{R^{\frac{3\eta + 1}{\eta}}}{1+3\eta}  = \frac{4\pi}{3} \frac{S}{1+3\eta} A^{-\frac{1}{\eta}}R^{\frac{3\eta + 1}{\eta}}  = \frac{1}{1+3\eta} S \left(\frac{4\pi}{3}R^3\right) \left(\frac{R}{A}\right)^{\frac{1}{\eta}}
\implies Q  = (1+3\eta)^{-1} S V t ,
which is equation (1) of the paper.

The number of supernova remnants that actually overlap a given point, P will be a Poisson distribution with mean, Q. This implies that the filling factor of the remnants is f = 1-e^{-Q}.

Using the rate of supernova events in our Galaxy, assuming that they expand until they reach pressure equilibrium with the ISM, and using some fiducial density and pressure for the ISM, McKee and Ostriker determine that the expected overlap factor, Q for supernova remnants would be larger than unity. The remnants therefore have a filling factor near one and the supernova bubbles would overlap before they could dissipate their energy. Since the interior of a supernova remnant is hot and has a low density, this implies that supernovae would readily create a hot, sparse phase of the ISM: a Hot Interstellar Medium.

What stops a Supernova?

  1. External Pressure, R_{\rm E}
    In the Sedov-Taylor phase, the internal pressure of the remnant drops as the supernova expands adiabatically and carries out PdV work on the ambient medium. Once the internal and external pressure equalise, the remnant will stop expanding. The maximum radius, R_{\rm E} that a remnant can expand to before it reaches pressure equalisation depends on the energy of the explosion, the ambient density and ambient pressure.
  2. Overlap, R_{\rm ov}
    Depending on the rate of supernova explosions, the blast waves of different remnants may run into each other before they have cooled. In this case, giant ionized bubbles will be formed in the ISM. Such objects are indeed seen in dense star-forming regions as H II superbubbles .
  3. Cloud-Shell collisions
    Once the cooling time drops below the age of the remnant, a thin shell of material forms at the edge (see stages of a supernova explosion). As the shell expands into the ISM, it runs into dense knots of molecular gas, which puncture the shell. The shell will be able to heal itself only if the typical distance between clouds, \lambda_{\rm w} multiplied by the ambient density, n_0 is comparable to or higher than the surface mass density in the shell. Otherwise the shell will slowly disintegrate as it passes through these dense clouds.
  4. Cooling
    As the supernova ages, it radiates away some of its energy and thereby loses steam. McKee and Ostriker divide radiative cooling losses into two kinds. 

    1. Cloud Crushing
      When the supernova blast wave shock crosses a small cloud of cold gas, radiative shocks are driven into the cloud, which dissipate some of the energy of the supernova shock.
    2. Direct Radiative losses
      In addition to expending energy on snuffing out clouds, a supernova remnant may also cool off by radiating energy directly. At the high post-shock temperatures inside the ejecta, most of the matter is ionized. Radiative cooling then takes place via thermal bremsstrahlung, which is highly inefficient at the low densities inside the remnant.

McKee and Ostriker’s view of the ISM

Fig. 1 - Cross section of a characteristic small cloud. Fig. 2 - Small scale structure of the ISM with a supernova blast wave sweeping through from the top right.

The authors view the ISM as being composed of numerous small (spherical!) clouds of molecular gas, embedded in a diffuse hot ISM (HIM). Each cloud has an ionized halo (the WIM) maintained by the interstellar UV background. Between the ionized halo and the cloud itself, they suggest the presence of a neutral zone heated by interstellar X-rays.

Cold Clouds (the Cold Neutral Medium)

Evaporation

In section II of their paper, the authors study the impact of evaporation of cold clouds on the evolution of supernova remnants. When the shock sweeps past a cloud, the cloud starts evaporating at a rate determined by the post shock temperature, T_{\rm h}, the cloud radius, a and an efficiency factor, \phi. This process of evaporation raises the density of the interior of the SNR. The authors assume that the medium behind the shock is at a constant density (!) and temperature and further that the magnetic fields inside the remnant are connected with the evaporating clouds (!), while the blast wave speed is proportional to the sound speed inside the SNR. Together with the principles of energy and mass conservation, the authors are able to determine the adiabatic evolution of the blast wave, R(t) and the overlap rate, Q(t) (equation 5 in the paper). Using a bremsstrahlung cooling function valid for 10^5\,{\rm K} < T < 4\times 10^7\,{\rm K}, appropriate for a young SNR, they determine the cooling time, the blast wave radius and velocity at the cooling time, and the density, temperature and pressure inside the remnant when it has reached its cooling time (equation 9 of the paper). The cooling time is important, since it marks the onset of shell-formation and modifies the dynamics of the interaction of the remnant with the ISM. This article is the first attempt at including the effect of the cold neutral medium on the propagation of supernova shocks.

Size Distribution

A summary of the power law size distribution for cold clouds used in McKee and Ostriker (1977)

In section IV of their paper, the authors discuss the size distribution of the clouds that form the cold neutral medium. They distinguish between the “true” sizes of the clouds from the sizes the clouds would have in the absence of ionizing radiation. They assume a power law distribution of cloud sizes, with N(a_0) {\rm d} a_0 = N_0 a_0^4 {\rm d}a_0. If a cloud gets too large, it will collapse under its own gravity. If it is too small, it will get obliterated by passing supernova shocks. With these guiding principles, the authors terminate the cloud size distribution above and below critical radii. With the evolution of supernovae in a clumpy ISM worked out in section II, the authors are able to work out the entire cloud size distribution that could be maintained in this environment (section V, equation 49).

The Warm Ionized Medium (WIM)

In this model, each (spherical) cold cloud is embedded in an ionized halo, produced by UV photons that come from B stars, supernova remnants, and central stars of planetary nebulae (i.e. white dwarfs). The authors fold in the ionization of the cold clouds in the determination of the radii and density of these ionized zones, and find that the CNM dominates over the WIM in terms of mass.

As a byproduct of this model, the authors predict that the CNM and WIM should be spatially correlated.

The Warm Neutral Medium (WNM)

The authors suggest that X-ray photons (with energy densities estimated from Chevalier’s (1974) calculation of energetics in supernova remnants) may penetrate through the WIM surrounding cold clouds and heat a layer of cold material to WIM temperatures (8000 K, in their model), without ionizing it. This was their Warm Neutral Medium (WNM), but they were unable to derive constraints on its properties or, indeed, to predict whether it would exist at all.

The Hot Ionized Medium (HIM)

Cox and Smith (1974) studied the impact of supernova explosions on the ISM by considering a uniform medium, and predicted that the explosions would create a network of “tunnels” of hot gas, which would be too diffuse to cool efficiently and would be supported by new supernova blast waves preferentially travelling through pre-existing tunnels. McKee and Ostriker expanded on that work, included the effects of cold clouds on the traversal of supernova shocks. They also inverted the geometry: with cold clouds embedded in a hot diffuse medium – instead of tunnels of hot gas in a cold medium.

Observational Evidence

McKee and Ostriker (1977) support their arguments for the presence of a Hot Interstellar Medium by pointing to some examples of observational evidence of diffuse hot gas in the Galaxy:

  • Soft X-Ray background
    In a rocket-borne experiment, Burstein et al (1977) observed \pi/2 steradian of the Galaxy and found a component of hot gas with a temperature, T > 10^{6.3}\,K based on thermal Bremsstrahlung.
  • O VI absorption lines
    Using the results of the Copernicus satellite of a spectroscopic study of nearby hot stars in the UV, Jenkins and Meloy (1974) confirmed – based on their narrow widths – that the O VI lines seen in stellar spectral arose in the ISM. They suggested that this indicated the presence of interstellar hot gas with a large filling factor.
  • Predictions

    1. Assuming that the supernovae expand until they reach pressure equilibrium with the ISM, the authors use mass and energy balance to predict the density (n_{\rm h}), pressure (\tilde{P}_{\rm h}) and temperature (T_{\rm h}) of the HIM. In their model, these values depend on the typical supernova energy, expansion speed, cooling efficiency, supernova occurrence rate, and packing fraction at the cooling radius. Since they assume that most of the phases are in approximate pressure equilibrium, \tilde{P}_{\rm h} is also the mean pressure of the ISM.
    2. They are able to tie the mean particle number density in the ISM to the supernova energy injection rate.
    3. Cloud stirring: supernova shock waves can also accelerate entire cold clouds, providing them with bulk kinetic energy. Given the supernova rate, they can balance this injection of energy against the losses due to inelastic cloud-cloud impacts, and thereby predict an equilibrium rms cloud velocity.Besides these, the authors also predict
    4. filling fractions for the various ISM phases,
    5. an expectation that the CNM and WIM phases must coexist spatially (except for very small core-less clouds), and
    6. that the Galaxy must be losing mass at a rate \sim 1\,M_{\odot}/{\rm yr} via a wind, due to injection of energy by supernovae into the hot gas in the Galactic halo.

    Unanswered Questions

    The following are some questions that are not addressed in this work:

    1. Where do cold clouds come from? What processes lead to their formation in the ISM? How are they stable against dissipation?
    2. What is the role played by star-formation? Giant Molecular Clouds? H II regions?
    3. The authors completely ignore an important store of energy and ionizing flux in the ISM: Cosmic Rays.

    What did they get right?

    • Supernovae are indeed important contributors to the dynamics of the ISM, and are indeed the dominant cause for the presence of a hot interstellar medium.

    What do we know better now?

    • The topology of the ISM as propounded by McKee and Ostriker was quite inaccurate. The WIM and CNM phases are not correlated (reference).
    • The pressures are larger than the typically adopted thermal pressures of the gas components (Heiles 2001, ASPC 231, 294H).
    • Magnetic fields play an important role in linking the various phases of the ISM together (Ibid).

    Despite their sweeping assumptions, ignorance of important contributors to the ISM and not quite getting the large scale structure of the ISM correct, McKee and Ostriker (1977) was a seminal work in the theory of the interstellar medium and (judging by the number of citations) has greatly impacted our understanding of the ISM today.

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    1. […] medium. For more on the different phases of the ISM, check out Tanmoy’s post about the 1977 McKee and Ostriker […]

    2. […] gas (Hot Ionized Medium): K. Shock heated from supernovae. Fills half the volume of the galaxy, and cools in about 1 […]

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