Harvard Astronomy 201b

Posts Tagged ‘ISM evidence/probes’

CHAPTER: Measuring States in the ISM

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

(updated for 2013)


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

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

SEDs

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

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

Spectra

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

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

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

A schematic of several structures in the ISM

Key

A: Dense molecular cloud with stars forming within

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

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

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

C: Supernova remnant

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

D: General diffuse ISM

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

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.

150 Foot Nebular Spectrograph

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

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.

Selected references (in order of appearance in this article)

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)

Struve and Elvey,”Emission Nebulosities in Cygnus and Cepheus”, 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)

ARTICLE: On the Dark Markings in the Sky

In Journal Club, Journal Club 2013 on February 8, 2013 at 2:46 pm

On the Dark Markings in the Sky by Edward E. Barnard (1919)

Summary by Hope Chen

Abstract

By examining photographic plates of various regions on the sky, Edward E. Barnard concluded in this paper that what he called “dark markings” were in fact due to the obscuration of nearby nebulae in most cases. This result had a significant impact on the debate regarding the size and the dimension of the Milky Way and also the research of the interstellar medium, particularly work by Vesto Slipher, Heber Curtis and Robert Trumpler. The publication of  Photographic Atlas of Selected Regions of the Milky Way after Barnard’s death, which included many of the regions mentioned in the paper, further provided a new method of doing astronomy research. In this paper and the Atlas, we are also able to see a paradigm very different from that of today.

It is now well-known that the interstellar medium causes extinction of light from background stars. However, think of a time when the infrared imaging was impossible, and the word “photon” meant nothing but a suspicious idea. Back in such a time in the second decade of the twentieth century, Edward Edison Barnard, by looking at hundreds of photographic plates, proposed an insightful idea that “starless” patches of the sky were dark because they are obscured by nearby nebulae. This idea not only built the foundation of the modern concept of the interstellar medium, but also helped astronomers figure out that the Universe extended so much farther beyond the Milky Way.

Young Astronomer and His Obsession of the Sky

In 1919, E. E. Barnard published this paper and raised the idea that what he called “dark markings” are mostly obscuration from nebulae close to us. The journey, however, started long before the publication of this paper. Born in Nashville, Tennessee in 1857, Barnard was not able to receive much formal education owing to poverty. His first interest, which became important for his later career, was in photography. He started working as a photographer’s assistant at the age of nine, and the work continued throughout most of his teenage years. He then developed an interest in astronomy, or rather, “star-gazing,” and would go watch the sky almost every night with his own telescope. He took courses in natural sciences at Vanderbilt University and started his professional career as an astronomer at the Lick Observatory in 1888. He helped build the Bruce Photographic Telescope at the Lick Observatory and there he started taking pictures of the sky on photographic plates. He then moved on to his career at the Yerkes Observatory at Chicago University and worked there until his death in 1922. (Introduction of the Atlas, Ref. 2)

One of the many plates in the Atlas including the region around Rho Ophiuchii, which was constantly mentioned in many of Barnard's works.

Fig. 1 One of the many plates in the Atlas including the region around Rho Ophiuchii, which was constantly mentioned in many of Barnard’s works. (Ref. 2)

Fig. 1 is one of the many plates taken at the Yerkes Observatory. It shows the region near Rho Ophiuchii, which was a region constantly and repetitively visited by Barnard and his telescope. Barnard noted in his description of this plate, “the [luminous] nebula itself is a beautiful object. With its outlying connections and the dark spot in which it is placed and the vacant lanes running to the East from it, … it gives every evidence that it obscures the stars beyond it.” Numerous similar comments spread throughout his descriptions of various regions covered in A Photographic Atlas of Selected Regions of the Milky Way (hereafter, the Atlas). Then finally in his 1919 paper, he concluded, “To me these are all conclusive evidence that masses of obscuring matter exist in space and are readily shown on photographs with the ordinary portrait lenses,” although “what the nature of this matter may be is quite another thing.” The publication of these plates in the Atlas (unfortunately after his death, put together by Miss Mary R. Calvert, who was Barnard’s assistant at the Yerkes Observatory and helped publish many of Barnard’s works after his death) also provided a new way of conducting astronomical research just as the World Wide Telescope does today. The Atlas for the first time allowed researchers to examine the image and the astronomical coordinates along with lists of dominant objects at the same time.

Except quoting Vesto Slipher’s work on spectrometry measurements of these nebulae, most of the evidences in Barnard’s paper seemed rather qualitative than quantitative. So, as of today’s standard, was the “evidence” really conclusive? Again, the question cannot be answered without knowing the limits of astronomical research at the time. Besides an immature understanding of the underlying physics, astronomers in the beginning of the twentieth century were limited by the lack of tools on both the observation and analysis fronts. Photographic plates as those in the Atlas were pretty much the most advanced imaging technique at the time, on which even a quantitative description of “brightness” was not easy, not to mention an estimation of the extinction of these “dark markings.” However, this being said, a very meaningful and somewhat “quantitative” assumption was drawn in Barnard’s paper: the field stars were more or less uniformly distributed. Barnard came to this assumption by looking at many different places, both in the galactic plane and off the plane, and observing the densities of field stars in these regions. Although numbers were not given in the paper, this was inherently similar to a star count study. Eventually, this assumption lead to what Barnard thought as the conclusive evidence of these dark markings being obscuring nebulae instead of “vacancies.” Considering the many technical limits at the time, while the paper might not seem to be scientific in today’s standard, this paper did pose a “conclusion” which was strong enough to sustain many of the more quantitative following examinations.

The “Great Debate”

Almost at the same time, perviously mentioned Vesto Slipher (working at the Lowell Observatory) began taking spectroscopic measurements of various clouds and tried to understand the constituents of these clouds. Although limited by the wavelength range and the knowledge of different radiative processes (the molecular transition line emission used largely in the research of the interstellar medium today was not observed until half a century later in 1970, by Robert Wilson, who, on a side note, also discovered the Cosmic Microwave Background), Slipher was able to determine the velocities of clusters by measuring the Doppler shifts and concluded that many of these clusters move at a faster rate than the escape velocity of the Milky Way (Fig. 2). This result, coupled with Barnard’s view of intervening nebulae, revolutionized the notion of the Universe in the 1920s.

The velocity measurements from spectroscopic observations done by Vesto Slipher.

Fig. 2 The velocity measurements from spectroscopic observations done by Vesto Slipher. (Ref. 3)

On April 26, 1920 (and in much of the 1920s), the “Great Debate” took place between Harlow Shapley (the Director of Harvard College Observatory at the time) and Curtis Heber (the Lick Observatory, 1902 to 1920). The general debate concerned the dimension of the Universe and the Milky Way, but the basic issue was simply whether distant “spiral nebulae” were small and lay within the Milky Way or whether they were large and independent galaxies. Besides the distance and the velocity measurements, which suffered from large uncertainties due to the technique available at the time, Curtis Heber was able to “win” the debate by claiming that dark lanes in the “Great Andromeda Nebula” resemble local dark clouds as those observed by Barnard (Fig. 3, taken in 1899). The result of the debate then sparked a large amount of work on “extragalactic astronomy” in the next two decades and was treated as the beginning of this particular research field.

The photographic plate of the "Great Andromeda Nebula" taken in 1988 by Isaac Roberts.

Fig. 3 The photographic plate of the “Great Andromeda Nebula” taken in 1988 by Isaac Roberts.

The Paper Finally Has a Plot

Then after the first three decades of the twentieth century, astronomers were finally equipped with a relatively more correct view of the Universe, the idea of photons and quantum theory. In 1930, Robert J. Trumpler (the namesake of the Trumpler Award) published his paper about reddening and reconfirmed the existence of local “dark nebulae.” Fig. 4 shows the famous plot in his paper which showed discrepancies between diameter distances and photometric distances of clusters. In the same paper, Trumpler also tried to categorize effects of the ISM on light from background stars, including what he called “selective absorption” or reddening as it is known today. This paper, together with many of Trumpler’s other papers, is one of the first systematic research results in understanding the properties of Barnard’s dark nebulae, which are now known under various names such as clouds, clumps, and filaments, in the interstellar medium.

Trumpler's measurements of diameter distances v. photometric distances for various clusters.

Fig. 4 Trumpler’s measurements of diameter distances v. photometric distances for various clusters.

Moral of the Story

As Alyssa said in class, it is often more beneficial than we thought to understand what astronomers knew and didn’t know at different periods of time and how we came to know what we see as common sense today, not only in the historically interesting sense but also in the sense of better understanding of various ideas. In this paper, Barnard demonstrated a paradigm which we may call unscientific today but made a huge leap into what later became the modern research field of the interstellar medium.

Selected References

  1. On the Dark Markings in the Sky, E. E. Barnard (1919)
  2. A Photographic Atlas of Selected Regions of the Milky Way, E. E. Barnard, compiled by Edwin B. Frost and Mary R. Calvert (1927)
  3. Spectrographic Observations of Nebulae, V. M. Slipher (1915)
  4. Absorption of Light in the Galactic System, R. J. Trumpler (1930)

CHAPTER: How do we know there is an ISM?

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

(updated for 2013)

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

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

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