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Technical Notes
LISTING OF Technical Notes
I. A GALAXY PRIMER 1. Distances 2. Numbers 3. Magnitudes 6. Galaxy Morphology -- Surface Brightness Distributions 9. Astronomical Photometry and Colors 10. Radiative Properties of Black Bodies 12. Star Groups 13. Phases of Interstellar Gas 14. Doppler Shifts 15. Reddening and Extinction by Dust 16. Dynamical Masses of Galaxies and Galaxy Clusters 17. The Jeans Criterion for Gravitational Instability
II. NEARBY GALAXIES 18. Sky Brightness and Light Pollution 19. Counting Stars 20. Radiative Powering of H II Regions 22. The Nearest Galaxies -- The Local Group 24. Starbursting and Interacting Galaxiesv 25. Tidal Action 26. Galaxies with Active Nuclei
III. OUR GALAXIAN UNIVERSE 28. Galaxy Clusters and Superclusters
EPILOGUE 30. The Drake Equation -- A Prospectus for Intelligent Life
Two units of distance are frequently used in the discussion of galaxies -- the light-year and the parsec. A light-year is the distance light travels in one year -- roughly 6 trillion miles (6 x 1012 miles) or 9.5 trillion kilometers (9.5 x 1012 km).2 A parsec (pc) is a peculiar unit of distance that is defined in terms of the apparent shifting (parallax) of stars across the heavens as the Earth revolves around the Sun. To visualize this shifting, try sticking your thumb in front of your face while fixing your gaze on the opposite wall. Look first with one eye open, then with the other eye. You will find that your thumb has shifted in position relative to objects on the opposite wall. Now, place your thumb at arm’s length and repeat the experiment. The "parallactic shift" of your thumb with respect to the background objects should be much less. Using this same triangulating principle, but with the orbiting Earth as the point of reference, astronomers measure the seasonal parallactic shifts of nearby stars with respect to the "fixed" background of more distant stars. More specifically, the distance to a star as measured in pc is reciprocal to that star’s parallax as measured in seconds of arc, ie. d(pc) = 1/p(arcsec), where the parallax angle (p) is one-half the total parallactic shift measured over 6 months, and where 1 arcsec = 1/3,600 of a degree. In other words, a star showing a parallax of 1 arcsec is 1 pc distant, while a star having a smaller parallax of 0.1 arcsec is 10 pc away. There are 3.26 light-years in a parsec (pc), 1,000 parsecs in a kiloparsec (kpc), and 1,000,000 (one million) parsecs in a megaparsec (Mpc). Until very recently, accurate parallactic distances had been measured for only a few hundred stars -- all within 25 pc (85 light-years) of us. With the successful deployment of the Hipparcos astrometric satellite from 1989 to 1993, we now have accurate distances to more than 7,000 stars extending out to 150 pc (500 light-years) radius from the Sun. This new bumper crop of well-fathomed stars has impacted many astronomical fields -- from the intrinsic luminosities of massive stars, to the luminosities & ages of globular cluster stars, to the distance scale and age of the Universe itself. To handle the vast ranges of distances, sizes, masses, and luminosities in the universe, astronomers must be adept at manipulating an equally large range of numbers. Towards these ends, "scientific notation" is the numbering system of choice. In the following table, examples of numbers are respectively expressed in words, decimal notation, and scientific notation. Table E1: Numerical Notations
Each number in scientific notation includes some power of ten (the exponent) along with a multiplicative coefficient (e.g. 6 x 1012). The exponent corresponds to the number of zeros that would be added to the coefficient, when expressing the number in decimal notation. Fractional numbers are expressed in similar fashion: 1/10 = 10-1, 2/10 = 2 x 10-1, 1/100 = 10-2, 2/100 = 2 x 10-2, 1/1,000 = 10-3, 1/1,000,000 = 10-6, and so forth. To add two numbers, one gets them into the same power of ten and then adds the coefficients. For example, 400 + 4,000 = (4 x 102) + (40 x 102) = 44 x 102. Subtraction works the same way. To multiply two numbers, add the exponents and multiply the coefficients. For example, 400 x 4,000 = (4 x 102) x (4 x 103) = 16 x 105. To divide two numbers, subtract the exponents and divide the coefficients. For example, 400 / 4,000 = (4 x 102) / (4 x 103) = 1 x 10-1. To raise a number to a power, multiply the exponent by that power and raise the coefficient to the same power. For example, (4,000)3 = (4 x 103)3 = 43 x 109 = 64 x 109. The magnitude system is an ancient and -- it turns out -- logarithmically compressed way of expressing the vast range of brightnesses that are encountered in astronomy. As such, it is similar to the Decibel system of quantifying different intensities of sound, and the Richter system of quantifying earthquake intensities. By convention, the difference between two apparent magnitudes is related to the ratio of brightnesses (or fluxes [f]) by m2 - m1 = -2.5 log (f2/f1), where the flux is often measured in units of ergs/second of power per square centimeter at the telescope (or equivalently, Watts/m2), and the minus sign indicates that the higher the flux is, the lower the corresponding magnitude becomes. For example, the brightest naked-eye stars are of 0th to 1st magnitude, while the faintest naked-eye stars are of 6th magnitude. From the above equation, one finds that a difference of 1 magnitude corresponds to a flux ratio of 1/2.512, and a difference of 5 magnitudes indicates a flux ratio of 1/(2.512)5, or 1/100. Counterparts of the above equation hold for expressing surface brightness (emitted flux per unit area of the source) in terms of magnitudes per unit area -- as is often used in studies of resolved objects such as galaxies -- and when dealing with intrinsic luminosities and their corresponding absolute magnitudes. In this latter case, one has M2 - M1 = -2.5 log (L2/L1), where the luminosity (L) is the intrinsic power of the source (often measured in erg/s), and the absolute magnitude (M) is arbitrarily set to zero for a star 10 parsecs away which has an apparent magnitude (m) of zero. Vega, the fiducial A0-type main-sequence star in the constellation of Lyra, comes very close to meeting this special circumstance (see Endnote 11). We can use the Sun as a handy reference when converting absolute magnitudes into intrinsic luminosities. According to the definition for magnitudes, one gets M(Source) - M(Sun) = -2.5 log {L(Source)/L(Sun)}, or L(Source)/L(Sun) = 10 -0.4{M(Source) - M(Sun)}. For example, a spiral galaxy with an absolute visual magnitude of MV = -20.15 mags can be compared with the Sun, where MV(Sun) = 4.85 magnitudes. According to the above relation, the spiral galaxy has a power output at visual wavelengths equal to 1010 (10 billion) Suns. The full relation between a source’s absolute magnitude (M) and apparent magnitude (m) critically depends on the distance to the source, such that m - M = 5 log (d/10), with the distance (d) expressed in parsecs. According to this system, the Sun has an absolute visual magnitude of MV = 4.85 mag, whereas its apparent magnitude as seen from Earth is a whopping mV = -26.72 mag. Turning the above relation around, one obtains the following expression for determining distance when both the apparent and absolute magnitudes are known ... d = 101+0.2(m - M). Because of its fundamental importance in determining distances, the quantity (m - M) is commonly known as the distance modulus. The concept of lookback time relies on the fact that light travels at a finite speed: c = v(light) = 186 thousand (1.86 x 105) miles/s, or 300 thousand (3 x 105) km/s. Consequently, light signals take a finite time to arrive from their sources. That means we see these sources as they were emitting a finite time ago. The farther away the source, the greater the lookback time. For example, we see the Moon as it was roughly a second ago, the Sun as it was 8 minutes ago, Jupiter as it was 30 minutes ago, Pluto as it was 5 hours ago, Alpha Centauri as it was 4 years ago, the Andromeda galaxy (M31) as it was 2.5 million years ago, and the most distant galaxies as they were roughly 5-15 billion years ago. In other words, the lookback time to the most distant known galaxies is about 5-15 billion years. Most nearby galaxies are bright enough to have been included in the nineteenth-century catalogue of nonstellar objects, assembled by J. L. E. Dreyer and called the New General Catalogue (hence the NGC numbers). The brightest of these galaxies are also listed in Charles Messier’s famous compendium of nebulous appearing objects, originally intended as a resource for 18th century comet watchers who might otherwise be fooled by these cometary imposters. For example, the Great Nebula in Andromeda is listed as NGC 224 in the New General Catalogue, and as M31 in Messier’s earlier catalogue. Faint galaxies and radio sources are usually named after the catalogue in which they first appear. For example, 3C 273, a quasar, is number 273 in the Third Cambridge Catalogue of Radio Sources. 6. Galaxy Morphology -- Surface Brightness Distributions Unlike the pointlike stars, galaxies appear extended in the sky. Galaxies in the Local Group span several degrees, while many of the "picture book" spirals beyond the Local Group subtend multiple arcminutes. Even the most remote galaxies in the Hubble Deep Field are resolved at the sub-arcsecond level. To describe the luminous shapes of galaxies, astronomers refer to the spatial distributions of surface brightness in these diverse realms. Surface Brightness The surface brightness (intensity) is a measure of radiant flux per unit area -- either in the plane of the sky or in the principle plane of the galaxy itself. It is often expressed with the symbol S or I (as used here) with units of (erg s-1 cm-2 arcsec-2) or (Watts m-2 steradian-1) on the sky. In the arcane system of magnitudes, the surface brightness is symbolized with m which has typical units of (mag arcsec-2). Although the flux from an unresolved source is critically dependent on that source’s distance, the surface brightness is independent of distance. The reason for this independence is that the inverse-square reduction in flux as a function of distance is exactly compensated by the increasing physical area of emission (per unit angular area on the sky). This nifty attribute means that one can directly translate the measured surface brightness on the sky into an actual luminosity per unit area at the galaxy itself. In frequently-used units of solar luminosities per square parsec, the conversion becomes I(LSun pc-2) = 1.34 x 1015 I(erg s-1 cm-2 arcsec-2). Radial Profiles of Surface Brightness One of the most powerful ways to quantify galaxy types is to measure and compare the radial distributions of surface brightness (intensity). Although the intensity is distance independent, the radius is not. For this reason, astronomers will sometimes refer to radial distances relative to some standard radius -- say the radius where the surface brightness falls to 25 mag/arcsec2 (the de Vaucouleurs radius). Other special radii depend on the central concentration of the light, as seen in the following formulations. There are several empirically derived relations for describing the radial distribution of intensities in elliptical galaxies. Hubble’s law is one of the most simple, with I(R) = Io/(1 + [R/Ro])2, where Io is the central intensity (measured in ergs/[s cm2 arcsec2] or L(Sun)/pc2, for example), and Ro is the projected radius where the intensity is 1/4 that in the center. At large radii, this relation reduces to the simple 1/R2 dependence discussed in Chapter 2. The de Vaucouleur "R1/4" law comes closest to fitting the radial profiles of elliptical galaxies over the greatest range of radii. It also has the advantage of yielding a finite total flux when integrated to infinite radius. It is basically an exponential relation with a 1/4 power in the exponent, ie. I(R) = Ie exp {-7.67(-[R/Re]1/4 - 1)}, where Re in this case is the radius inside of which half the light is contained, and Ie is the corresponding intensity at this radius. With this definition, the central intensity is 2140 Ie -- very bright indeed. The de Vaucouleurs law turns out to be part of a family of exponential relations that were first formalized by J. L. Sersic in the 1960’s. In its most generic form, the surface brightness is related to the projected radius by I(R) = Io exp (-[R/Ro]1/n). For n = 1, one has the simple exponential relation seen in galaxian disks. For n = 4, one has something like the de Vaucouleurs "R1/4" relation, and for higher n, one obtains power-law dependences akin to that formulated by Hubble. Between n = 1 and n = 2, good fits are obtained for the bulges of many galaxies -- suggesting their hybrid nature. All of these empirical fits to the luminous forms of galaxies are just that -- formal expressions that can only hint at the dynamics underlying these vast and complex systems. For physicists, the temperature of a gas is specifically related to the motion (or kinetic energy) of the gas particles by the following relation K.E. = (1/2) m <v2> = (3/2) k T, where K.E. is the kinetic energy, m is the mass of each particle in the gas, v is its velocity, the brackets <> denote an averaging of the particle speeds, k is Boltzmann’s constant (named after Ludwig Boltzmann [1844 - 1906] who first successfully related the kinetic statistics of gas particles to the thermodynamic properties of the macroscopic gas; k = 1.38 x 10-16 erg/Kelvin, in centimeter-gram-second (CGS) units), and T is the absolute temperature in units of Kelvins. At a constant (isothermal) temperature, the gas will have a known average kinetic energy per particle. By likening the stars in an elliptical galaxy to the particles of an isothermal gas, one can relate the kinetic and gravitational energies, and so derive a self-gravitating "swarm" of stars whose density falls off as 1/r2. Let the swarm be in virial equilibrium, such that the gravitational energy is twice the kinetic energy and is of opposite sign, G. E. + 2 K.E. = 0. By comparison, a system on the brink of breaking up would have the two opposing energies match one another. Expressing the virial relation in terms of the interior mass M and test mass m yields -GmM/r + 2 (m<v2>/2) = 0, or G M = <v2> r, where <v2> is the mean square velocity of the "isothermal" swarm, and G is Newton’s constant of universal gravitation (G = 6.67 x 10-8 dyne cm2 g-1, in CGS units). In differential form, this becomes G dM = <v2> dr. Assuming spherical symmetry, the differential mass dM can be expressed in terms of the density r and radius r, dM = r(r) (4pr2) dr, where (4pr2) dr is the volume of the shell containing the differential mass. The virial relation then reduces to G r(r) (4pr2) dr = <v2> dr so that the solution for density becomes r(r) = <v2> / G (4pr2). Integration of this 1/r2 solution along the line of sight yields the projected density distribution s(R) which goes as 1/R. Assumption of a constant mass-to-light M/L ratio throughout the model galaxy results in a surface brightness profile I(R) that has the same 1/R dependence. One problem with this sort of model is that the surface brightness rises to infinity at zero radius. To ameliorate this undesirable effect, Ivan King generated in the 1960’s a set of quasi-thermal models whose central densities and corresponding intensities saturate at a finite level. These so-called King models successfully describe the surface brightness distributions observed in globular clusters -- thus indicating that these systems are pretty much thermalized and hence dynamically "relaxed." Elliptical galaxies and the bulges of spiral galaxies often have steeper, less saturated profiles, however, which are better modeled with the Hubble and de Vaucouleur relations and which may indicate other, less "thermalized," dynamics at work (see Endnote 6 and Chapter 2). To illustrate the effects of differential rotation, consider a disk galaxy with a typically "flat" rotation curve, such that v(R) = constant = vo. The orbital frequency is then the orbital velocity divided by the orbital circumference, such that f(R) = 1/T(R) = v(R)/2pR = vo/2pR, where f(R) is the orbital frequency, and T(R) is the orbital period. Because the orbital frequency declines with radius, the disk is not rotating like a solid body (such as a record or CD) but is in a state of differential rotation where the inner parts shear past the outer parts. Given a material arm connecting any two radii in a galaxy, the total number of arm windings is obtained from the difference of orbital frequencies times the age of the system (t), whereby # windings = t {f(R2) - f(R1)} = tvo/2p {(1/R2 - 1/R1)}. For the Sun’s orbit in the Milky Way, we have vo = 220 km/s and R1 = 8.5 kpc (2.63 x 1017 km). If a material arm originally connected the Solar Neighborhood with the molecular ring at R2 = 5.1 kpc (1.58 x 1017 km), the differential rotation over a timescale of 1010 years (3.15 x 1017 s) would have wound this arm up 28 times. The fact that spiral arms are observed to wind up no more than one or two times argues strongly for the arms being either transitory or non-material (ie. density wavefronts). 9. Astronomical Photometry and Colors The light from stars and stellar systems is typically measured through filters of known bandpasses. By using standardized filters, one can compare photometric measurements of different sources -- independent of the observer and observatory. The following table summarizes the popular Johnson system of wide-band filters. Other frequently used wide-band filter systems include the UBVRI Mould system used at Kitt Peak National Observatory, the uvbyb Stromgren system configured to avoid specific emission lines, and the Hubble Space Telescope’s own system. At optical wavelengths, filter bandpasses are often expressed in nanometers (1 nm = 10-9 m) or in Angstroms (1 Å = 10-10 m). At the longer infrared wavelengths, microns (mm) are the preferred units (1 mm = 10-6 m). The bandwidth of a filter is usually expressed as the full-width at half-maximum (FWHM) of the filter’s transmissivity function. Wide-band filters such as those tabulated below are used to characterize the continuum emission from an astronomical source in terms of its overall brightness and color. Narrow-band filters (bandwidths less than 10 nm) are used for the isolation of particular spectral features such as emission lines and discontinuities ("jumps") in the stellar continuum emission. Table E2: Johnson Filter System
Astronomical colors involve flux measurements through two different filters, and are often expressed as a difference between the two measured magnitudes, whereby m(l2) - m(l1) = -2.5 log {f(l2)/f(l1)}. For example, the (B - V) color index is obtained from (B - V) = m(B) - m(V) = -2.5 log {f(B)/f(V)}, where f(B) and f(V) are respective measurements of the flux per unit wavelength through the B and V-band filters. The characteristic (B - V) colors of main-sequence stars are tabulated in the Endnote 11. 10. Radiative Properties of Black Bodies Black Bodies are perfect radiators, such that their spectral energy distributions (SEDs) depend only on the body’s surface temperature. The flux per unit frequency interval (dn), unit area of emitting surface (dS), and unit solid angle of direction (dw) is known as the Planck function, after the German physicist Max Planck (1858 -- 1947) who first explained the observed emission from hot bodies in terms of quantum processes. It is expressed as follows Bn(T) = {2hn3/c2} / {ehn/kT - 1}, where (h) is Planck’s constant (h = 6.62 x 10-27 erg-s), (k) is Boltzmann’s constant (k = 1.38 x 10-16 erg/Kelvin), (c) is the speed of light (c = 3.00 x 1010 cm/s), and (Bn) is in units of (ergs/[s cm2 Hertz steradian]). Sometimes it is more convenient to re-write this function in terms of wavelength (l), where l = c/n, and |dl| = cdn/n2, so that Bl(T) = {2hc2/l5} / {ehc/lkT - 1}. The maximum of this function occurs where its slope goes to zero, ie. where dBl/dl = 0. The corresponding wavelength of peak emissivity is thus found to be inversely proportional to the temperature, whereby l(peak) = C/T. For instance, a doubling of the temperature results in the wavelength of peak emissivity being reduced by 1/2 (see Figure 3.1). This relation is known as Wein’s Displacement Law. The constant C depends on the particular unit of wavelength. In units of meters, C = 2.898 x 10-3 K-m; in cm, C = 2.898 x 10-1 K-cm; in microns, C = 2.898 x 103 K-mm; in nanometers, C = 2.898 x 106 K-nm; and in Angstroms, C = 2.898 x 107 K-Å. Integrating the Planck function over all possible solid angles (in steradians) outward from the unit area of emitting surface produces what is known as the monochromatic flux Fn(T) = pBn(T), or Fl(T) = pBl(T). Further integrating the monochromatic flux over all frequencies (or wavelengths) produces the so-called bolometric surface flux F(T) = sT4, where s is the Stefan-Boltzmann constant (s = 5.67 x 10-5 ergs/[s cm2 Kelvin4]). Note the high power in the exponent of the temperature, and correspondingly high sensitivity of the surface flux to any changes in the temperature. For example, if the temperature doubles, the surface flux will increase by a factor of 24 = 16! By assuming spherical symmetry (as in a star), one can integrate over the total spherical surface area and so obtain the total (bolometric) luminosity L(R,T) = (4pR2) sT4. Simple comparisons between stars can be made by assuming black-body emissivities and taking ratios, such that L2/L1 = (R2/R1)2 (T2/T1)4. For example, the red supergiant star Betelgeuse is about 800 times larger than the Sun, but with a surface temperature that is roughly half that of the Sun. Its bolometric luminosity is then approximately (800)2 (1/2)4 = 40,000 times greater than the Sun’s total power output. The following table summarizes the basic properties of main sequence stars. This table is based on having established reliable distances to a large enough number of representative stars -- no easy task. The recent results of the Hipparcos astrometric satellite has helped tremendously in this regard (see Endnote 1).
Table E4: Main Sequence Stellar Properties
Notes to table: (1) Spectral type, based on spectral absorption-line features. (2) (B - V) color index in magnitudes. (3) Surface effective temperature in Kelvins, assuming a blackbody emissivity spectrum. (4) Absolute visual magnitude, based on apparent magnitude mV and distance d (see Technical Notes 1 and 3). (5) Bolometric (total) luminosity, based on absolute visual magnitude MV and bolometric correction B.C., which in turn is based on the stellar surface temperature. Lbol is expressed in units of solar luminosities {Lbol(Sun) = 4 x 1033 ergs/s}, such that L/L(Sun) = 10-0.4{M - M(Sun)}, where the absolute bolometric magnitude of the Sun is Mbol(Sun) = 4.77 mag. (6) Mass in units of solar masses {M(Sun) = 2 x 1033 g}, based (mostly) on the orbital behavior of stars in binary star systems. For the highest mass stars {M > 20 M(Sun)}, nearby (resolvable) binary systems are lacking, and so stellar models are used. (7) Total main-sequence lifetime, based on theoretical mass-lifetime relations.
From this table, empirical relations can be drawn. Most notably for main sequence stars, the mass and luminosity are directly related by power laws L/L(Sun) = {M/M(Sun)}n For intermediate-to-high masses of 0.5 M(Sun) < M < 20 M(Sun), the exponent n @ 3.5. A crude idea of the total stellar lifetime can be obtained by dividing the "fuel" (M) by the available "fire" (L). More specifically, t/t(Sun) = {M/M(Sun))/(L/L(Sun)} which reduces to t/t(Sun) = (M/M(Sun)}1-n. Therefore, the stellar lifetime goes as t @ 1010 {M/M(Sun)}-2.5 years, a reasonable approximation to what the more sophisticated stellar models indicate. For example, a 10 solar-mass B3 main-sequence star has a predicted lifetime that is only 30 million years, while a 0.7 solar-mass K5 star should last more than 20 billion years. There are basically three kinds of star groups that have been found in galaxies. Stellar associations are loose, young groupings of recently formed stars, conspicuous because of their very high luminosity and temperature. They are typically larger than 100 light years across. Open clusters are smaller, more compact groups, relatively stable and with a variety of ages. Globular clusters are large, with total diameters of 100 to 300 light years, populous, containing 104 to 106 stars, and bright, with luminosities of 102 to 104 Suns. In our Milky Way galaxy, they are uniformly very old, generally including the oldest known stars known. In the neighboring Magellanic Clouds, however, a much wider range of globular cluster ages is found. 13. Phases of Interstellar Gas Up until the 1970's, the interstellar medium was thought to contain two phases of interstellar gas. The cool atomic hydrogen gas, first detected in the 1950's, was the densest phase. It had a temperature of about 100K and occurred in massive clouds. The warm ionized phase was 100 times hotter and 100 times less dense, comprising a tenuous intercloud medium. The resulting energy densities (and pressures) of the two phases were in rough equilibrium. Moreover, sufficient compression of the warm tenuous gas could drive thermal instabilities that would result in the gas switching its phase to the cool atomic state. This "two-phase" model provided one of the first cogent galactic ecosystems, in which clouds and subsequent star formation could develop in regions of gas compression (e.g. spiral arms). Since the 1970's, we have learned that the interstellar medium contains a far more complex stew of gaseous phases. The following table summarizes these various phases. The tabulated temperatures and densities are rounded off to the nearest power of ten. That is because these quantities vary by factors of 2--3 within each phase, and so any further specification would be more confusing than accurate. The product (nT) is directly proportional to the thermal energy density (u = nkT), where k is Boltzmann's constant. The thermal energy density also corresponds to the thermal gas pressure. From the table, we see that the cold molecular, cool atomic, warm atomic, warm diffuse ionized and hot coronal phases are in rough pressure equilibrium. Some evidence exists to support the idea that transitions can occur between some of these phases -- driven by compressive dynamics along with radiative cooling . By contrast, the H II regions and supernova remnants (SNRs) are overpressurized -- indicating expansive dynamics. The high-density molecular phase also appears overpressurized. However, gravitational binding plays an important role at these densities, thus preventing any expansion.
Table E4: Gaseous Phases
As Christian Doppler showed in the early 1800’s, the line-of-sight motion between a source of propagating waves and an observer ends up shifting the observed wavelength by a predictable amount. This so-called Doppler shifting applies to all types of propagating energy that can be described in terms of waves -- from the siren wail of a speeding ambulance to the nebular line emission from a swirling galaxy. When the source and observer are in relative motion toward one another, the wavefronts emanating from the source scrunch up in the direction of motion, shortening the wavelength and so producing a blueward shift, or "blueshift." When the source and observer are in relative motion away from one another, the emanating wavefronts are stretched out in the receding direction, producing a redward shift in wavelength, or "redshift." The amount of shifting is denoted by z, and is directly proportional to the relative velocity along the line of sight between source and observer. This is quantified as z = Dl/lo = (l - lo)/lo = v/c, where l and lo are, respectively, the observed and emitted wavelengths, v is the component of velocity along the line of sight, and c is the speed of the propagating wave. For light in a vacuum, c = 3 x 1010 cm/s. By convention, approaching motions have negative velocities and hence negative (blue) shifts in wavelength. Receding motions have positive velocities and positive (red) shifts. The shifting of frequency n has the same dependence but of opposite sign, ie. Dn/n = -v/c. When the line-of-sight velocity is a significant fraction of light speed, relativistic time-dilation effects alter the relation to z = Dl/lo = (1 + v/c)/(1 - v2/c2)1/2. Unlike the non-relativistic relation, the redshift z can exceed unity without the relative velocity surpassing light speed and so violating the fundamental premise of Einstein’s theory of special relativity. The redshifts of galaxies are often discussed in terms of Doppler recession velocities. Indeed, the Hubble constant Ho that describes the expansion of the galaxian Universe is most commonly expressed as Ho = v/d, where v is found from the redshift z. However, the cosmological redshifts of galaxies are more closely related to the scale of the Universe than to any measurements of velocity. As our space-time expands, all propagating light waves stretch accordingly. The interpretation of galaxian redshifts as Doppler recession velocities no longer makes sense, because the galaxies are not moving relative to our space-time. It is our space-time that is expanding. 15. Reddening and Extinction by Dust Microscopic grains of cosmic dust interact with light in three basic ways. They absorb light with an efficiency that increases at shorter wavelengths. They scatter light with a similar wavelength-dependent efficiency. And they emit light with a spectral energy distribution that depends on their temperature. Here, we will concentrate on their absorbing properties and consequences. The most directly observable consequence of cosmic dust is the extinction of starlight that is produced by absorbing dust along the line of sight to the star. In the magnitude system (see Endnote 3), the extinction A(l) is defined as follows A(l) = m(l) - mo(l) = -2.5 log {f(l)/fo(l)}, where mo and fo respectively refer to the apparent magnitude and flux in the absence of extinction. Solving for the observed flux yields f(l) = fo(l) 10-0.4A(l), which is the same as f(l) = fo(l) e-t(l), where the extinction A(l) and optical depth t(l) are related by A(l) = 1.086 t(l). The optical depth refers to the actual amount of dust along the line of sight, such that t(l) = k(l) Nd, where k(l) is the absorption efficiency of the dust grains, and Nd is the column density of dust along the line of sight (in units of grains per unit projected area). Extinctions are determined from observations, while optical depths are typically derived from theoretical models. So it is handy to know how these two measures of dust are related. The dependence of extinction on wavelength varies with galactic locale. Key influences include the abundance of heavy elements (the metallicity) necessary to make dust grains, the density of the clouds containing the dust, and the intensity of ultraviolet light from hot stars. All of these factors can change the distribution of grain sizes as well as the grain geometries and chemical compositions. In the Milky Way, the average extinction is more or less inversely proportional to the wavelength, ie. A(l2)/A(l1) = t(l2)/t(l1) = k(l2)/k(l1) = l1/l2, with the biggest exception occurring near the ultraviolet wavelength of 215 nm, where carbonaceous grains are thought to produce the excess absorption that is observed. Curiously, this UV absorption feature is not observed in the Magellanic Clouds, nor in several other galaxies, where measurements of sufficient accuracy have been made. Because the wavelength-dependent extinction passes the redder light at the expense of the blue light, it is often termed reddening. In the magnitude system, it is expressed as a color excess, where E(l2 - l1) = A(l2) - A(l1). For example, through the Johnson blue (B) and visible (V) filters, the color excess goes as E(B - V) = A(B) - A(V). Determinations of the E(B - V) color excess from observations of nearby star clusters have shown that the color excess (reddening) is directly proportional to the total amount of visual extinction. This relationship is simply expressed as A(V) = RV E(B - V), where the constant of selective extinction RV has an average value of 3.2 in our Milky Way galaxy. Based on this relationship, one can obtain a reasonable estimate of a star’s visual extinction (and corresponding dust column density to that star) by measuring the reddening of the star’s light. For star clusters, this is often done by plotting the (B - V) colors with respect to the (U - B) colors of the stars and comparing the resulting distribution with the predicted color-color "tracks" of main-sequence and giant stars. By measuring the offset in the (B - V) and (U - B) colors relative to the theoretical tracks, one obtains a good estimate of the E(B - V) reddening and corresponding visual extinction A(V). Another method for determining the extinction to a source is by measuring the hydrogen line emission from the nebulae associated with hot stars (H II regions). The key is to measure two or more different hydrogen lines from the same source, and then to compare the resulting line ratios with those predicted by atomic theory. For example, in an H II region with a typical nebular temperature of 104 K, the nominal ratio of fluxes between the Balmer series Ha line at 656.3 nm wavelength and the shorter wavelength Hb (486.1 nm) line is fo(Ha)/fo(Hb) = 2.86. Measurements of higher flux ratios would then indicate the presence of reddening, whose quantification would provide the ingredients necessary to compute the nebular extinction. All of these methods for determining and compensating for the extinction rendered by dust assume that the dust is foreground to the source and is uniformly distributed -- as in a hazy slab. However, nature is often far more complicated and interesting than our naive assumptions allow. Mixing the emitting sources with the absorbing dust, or altering the geometry of the slab into a shell, or merely organizing the dust into clumps can have varying effects on the emergent radiation which are difficult to model. Sometimes, we find that our estimates of visual extinction based on hydrogen-line ratios systematically increase when we use lines of greater wavelength (e.g. the infrared Brackett series Bra and Brg lines at 4.05 mm and 2.16 mm, respectively). This behavior can be understood, if the sources and dust are co-mixed -- producing a "skin-depth" effect that is wavelength dependent. Simply put, the longer-wavelength emission is seen to greater depths in a dense emitting and absorbing cloud, so we sample more dust and measure greater extinctions at these wavelengths. That is why longer-wavelength infrared and radio observations are often preferred in studies of especially dusty systems. Other times, we find that our stellar and nebular measures of visual extinction towards the same source can differ. Most likely, geometric factors are to blame. In young star-forming regions, the stars are often more localized than the gas. Moreover, the stellar winds can evacuate their immediate surroundings -- reducing the extinction toward them compared to that found in the neighboring nebulae. In older stellar systems, the gas is frequently more localized than the stars -- producing entirely different effects. Given the many possible arrangements of stars, gas, and dust in galaxies, we must regard our estimates of reddening and extinction with a good bit of caution. 16. Dynamical Masses of Galaxies and Galaxy Clusters The motions of galaxies are governed by the gravitation that binds each galaxy to itself and this same galaxy to its neighbors. The gravitation, in turn, depends on the total mass that is involved. Therefore, galaxian motions provide critical measures of the masses in galaxies and in clusters of galaxies. As in Endnote 7, we can invoke the state of virial equilibrium, to relate the gravitational and kinetic energies and so derive the dynamical mass. This important relationship was obtained by Rudolf Clausius in 1870 from a statistical analysis of an "ensemble" containing many particles that are gravitationally interacting. The statistical ensemble is also dynamically stable, such that it is neither expanding nor contracting. By averaging the gravitational and kinetic energies of the many particles over all time, Clausius showed that the average gravitational energy (G.E.) of each particle must be twice the average kinetic energy (K.E.) and of opposite sign, ie. <G.E.> + 2 < K.E.> = 0, where the brackets denote the averaging. For comparison, an ensemble near escape velocity would have the two energies matching one another. The beauty of the virial relation is that it does not depend on the specific orbits of the particles. However, it is helpful to first consider the case of particles in circular co-planar orbits, as is seen in disk galaxies. In this special case, the virial relation reduces to -GMm/R + 2 (mv2/2) = 0, which can be re-written as GMm/R2 = mv2/R, where R is the orbital radius, M is the gravitating mass interior to the test particle, m is the mass of the test particle itself, and v is the rotation speed. To the discerning eye, this latter relation is none other than the equality between the gravitational and centripetal forces that prevails when gravitating particles are in circular orbit about some larger mass. Solving for the dominant mass yields M(R) = v2R/G. Here, the dominant mass is assumed to be concentrated to a small region compared to the radius (R) of the orbiting test particle. When dealing with gravitating masses that are not pointlike but are spatially distributed, the above solution must be altered according to the solution for the gravitational energy. For example, a spherically distributed mass of constant density has a gravitational energy of G.E. = -(3/5) GM2/R, and so the solution for the virial mass is M = (5/3) v2R/G. Regardless of these details, the basic solution shows that for a "flat" rotation curve, with v = constant, the mass increases linearly with radius. This is what we find in spiral galaxies, prompting our concerns over the inferred dark matter at large radii. The stars in elliptical galaxies and the galaxies in clusters do not stick to co-planar circular orbits. However, virial masses can be derived from the measured velocity dispersions. In this more generalized case, the virial relation has the following form -[C]GMm/R + 2 (m<v2>/2) = 0, where the constant [C] (of order unity) depends on the modeled density distribution, and the velocity dispersion <v2>1/2 is measured from the Doppler-broadened spectral lines. Again, the solution for mass has the same basic dependence, but with the velocity dispersion substituted for the rotation speed. For example, in spherical galaxies, the mass contained within the isophote that encloses half the total light has the specific solution of M = 3 <v2>Re/G, where Re is the half-light radius. The masses of galaxy clusters can be modeled in similar fashion, where the velocity dispersion is obtained from the measured radial velocity of each galaxy. The virial relation is also used to interpret the X-ray observations of galaxy clusters in terms of the underlying masses. From the X-ray spectrum, it is possible to determine the temperature of the emitting particles in the intracluster gas. This temperature, in turn, is related to the velocity dispersion of the particles by (1/2) m <v2> = (3/2) k T, where k is Boltzmann’s constant (1.38 x 10-16 erg/Kelvin). The virial relation then becomes -GMm/r + 2 (3/2) k T = 0, which reduces to the following solution for mass M(R) = 3kTR/GmH give or take a constant of order unity (where mH is the mass of the hydrogen nucleus, the dominant particle in the gas). For example, an X-ray emitting cluster with a gas temperature of 107K and a radius of 10 Mpc (3.1 x 1025 cm) would have a virial mass of 1.13 x 1048 grams, or 5.76 x 1014 solar masses! Similar cluster masses are obtained from the velocity dispersions of the cluster members themselves, thus corroborating the X-ray technique. The corresponding mass-to-light ratios are of order 100--300, further indicating copious amounts of dark matter. 17. The Jeans Criterion for Gravitational Instability In his original derivation of the conditions for gravitational instability, Sir James Jeans (1877 -- 1946) considered an infinite slab of gravitating matter that is perturbed by a sound wave. The resulting solution yielded a minimum wavelength, above which the matter would be unstable to gravitational collapse. This minimum wavelength is commonly known as the Jeans length. We can achieve the same ends by simply considering the gravitational and kinetic energies that are involved. For gravitational collapse to occur, the gravitational energy must exceed the kinetic energy, ie. -G.E. > K.E. For simplicity, we can consider the gravitational energy of a hydrogen atom in a spherical cloud of mass M, and assume that the kinetic energy is dominated by the thermal motions. In this case, the above inequality can be written as GMmH/R > (3/2) kT, where mH is the mass of the hydrogen atom. Rewriting this relation in terms of the density r (assumed constant throughout) yields Gr(4p/3)R2mH > (3/2) kT, whose solution for the gravitationally unstable radius is R > RJ = (9/8p)1/2 (kT/GrmH)1/2, where RJ is the Jeans radius, the minimum radius above which the cloud is unstable to gravitational collapse. Working through the various constants yields RJ(cm) = 2.1 x 107 (T/r)1/2 or RJ(pc) = 6.8 x 10-12 (T/r)1/2, where T is in Kelvins and r is in grams/cm3. The corresponding Jeans mass is MJ = (4p/3) RJ3 r, which in terms of temperature and density becomes MJ = (4p/3) (9/8p)3/2 (kT/GmH)3/2 r-1/2. Doing the arithmetic yields MJ(grams) = 3.9 x 1022 T3/2 r-1/2 or MJ/M(Sun) = 2.0 x 10-11 T3/2 r-1/2. The Jeans criterion for gravitational instability is often invoked to estimate the proto-galactic sizes and masses that first emerged following the epoch of decoupling, some 300 thousand years after the Big Bang (see Chapters 5, 15, and 16). The temperature then was about 3,000K, and the density was roughly 10-21 grams/cm3. The corresponding Jeans radius would have been about 12 pc, and the Jeans mass approximately 105 solar masses. Therefore, the Jeans criterion explains the apparent lack of galaxies with masses less than 105 solar masses. What the Jeans formulation for gravitational instabilities does not do is predict (or preclude) the existence of much larger galaxies (such as our Milky Way galaxy), nor does it explain the apparent upper limit of 1013 solar masses that is evident among galaxies in the local Universe. Perhaps the Jeans criterion played an important role at even earlier and warmer epochs, driving higher-mass instabilities in the plasma that did not get completely damped by the intense radiation field. Alternatively, the currently observed range of galaxy masses may have resulted from a rapid aggregation of smaller parcels shortly after the initial instabilities took hold (see Chapters 5 and 16). 18. Sky Brightness and Light Pollution The night sky has many sources of illumination -- beginning with the background light from distant galaxies, the starlight scattered off dust in the Milky Way, the so-called zodiacal light from sunlight scattered off dust in the Solar System, and the auroral glow from solar particles bombarding the Earth’s upper atmosphere. The Earth’s lower atmosphere -- even at its clearest and driest -- scatters whatever moonlight may be present, further brightening the night sky. During the dark of the Moon, the most remote groundbased observatories have typical sky brightness levels of 22 magnitudes per square arcsecond (22 mag/arcsec2) at V-band and 23 mag/arcsec2 at B-band. These values fluctuate somewhat, as the degree of solar activity and the amount of volcanic dust in the Earth’s atmosphere varies. At light-polluted sites, the sky brightness levels are dramatically higher. For example, Palomar Observatory (near San Diego) is brighter by 0.5 mag/arcsec2, Lick Observatory (near San José) is brighter by 1.5 mag/arcsec2, and Mount Wilson Observatory (near Los Angeles) is a whopping 2.5 mag/arcsec2 brighter -- amounting to a full 10-fold increase in illumination as measured in physical units of ergs/[s cm2 arcsec2]. The effects of urban light pollution is to decrease the numbers of stars that can be discerned with the naked eye from about 4,000 to less than a few hundred. Diffuse features in the night sky -- such as the Milky Way, zodiacal light, and aurorae -- are completely overwhelmed by the artificial illumination. For professional astronomers, the ability to detect and measure faint sources is severely curtailed at light-polluted sites -- thus limiting the research that is possible. Woeful circumstances, indeed. Tears notwithstanding, we are heartened to report that concerted efforts to reduce light pollution are beginning to show some successes. Through more intelligent and economical lighting methods that ensure public safety, several municipalities (Tucson and San Diego included) have begun to take back the night for all to enjoy. Proper shielding of street lamps, in particular, has done wonders in reducing light pollution and light trespass. These so-called full-cutoff lighting fixtures direct the light where it is intended, thus maximizing energy efficiency while minimizing wasteful illumination of the sky. For more information on how you can help encourage better lighting practices in your own locale, you can contact the International Dark Sky Association (IDA), whose website is (http://www.darksky.org/), e-mail address is (ida@darksky.org), and postal address is (3225 N. First Avenue, Tucson, AZ 85719). The arduous task of counting stars in the sky has yielded up some remarkable findings. These include the types of stars that dominate the mass of a galaxy, the types that dominate its light, and the overall spatial distributions of these various types. Before recognizing the important effects of interstellar dust, however, astronomers were seriously misled by their counts. Starcounts in the Galaxy: the Kapteyn Universe The basic job at hand is to measure distances to as many stars as possible. Once obtained, the distance of each star enables its spatial position and luminosity to be determined. The star’s spectral type can be determined from its visible-light spectrum or -- more readily -- from its color. Often, the star’s spectral type and apparent magnitude are compared with its expected absolute magnitude to calibrate its distance. When all is said and done, the astronomer should have a complete set of directions and distances for each bin of stellar luminosity and spectral type. From this database, luminosity functions and type-dependent spatial distributions can be derived. The Dutch astronomer Jacobus Kapteyn (1851 -- 1922) was the first to carry out all these steps as part of his massive survey of the northern sky. The resulting spatial distribution of stellar densities -- then the best representation of the stellar cosmos --became known as the Kapteyn universe. It showed a rather thick disk that was centered on the Sun. The disk measured about 55,400 light-years across and 11,100 light-years thick near the Sun -- tapering to negligible thickness near its extremity. Unfortunately for Kapteyn, the obscuring effects of interstellar dust had not yet been fully recognized. The deep photographs by Edward E. Barnard in the 1890’s gave the first telling indications of dark filamentary patches in the Milky Way, whose dusty nature he correctly interpreted. These morphological clues were followed in the 1920’s by Robert J. Trumpler’s study of open star clusters. Trumpler found that the relationship between a cluster’s apparent brightness and its angular size was not as predicted from the cluster’s inferred distance. That is because the intervening dust had introduced a serious error in the determination of distance. We can see how obscuration by dust introduces such errors by considering the basic distance modulus relationship between apparent and absolute magnitudes ... m - M = 5 log (d/10). Now, consider the effect of dust on the observed apparent magnitude ... m = mo + A, where mo is the apparent magnitude in the absence of obscuration, and A is the obscuration, or extinction. The distance modulus relation then becomes mo - M = m - M - A = 5 log (d/10). Solving for the distance yields d = 101+0.2(m - M - A). Given an observed apparent magnitude and independently determined absolute magnitude, a measured extinction is seen to decrease the true distance by do/d = 10-0.2A. For example, including an extinction of 1.5 magnitudes reduces the derived distance by 1/2. Turning this argument around, the reason why Trumpler’s clusters did not decrease in angular size as much as was predicted from their diminishing brightnesses is because the brightness-based distances were erroneously larger than would be reckoned in the presence of obscuration. Similarly, Kapteyn’s stars had distances that were systematically overestimated, the error increasing at greater distances (and corresponding column densities of dust). With erroneously greater distances being estimated for the fainter stars, Kapteyn’s universe quickly thinned out away from the Sun -- thus producing his heliocentric cosmos. This behavior can be quantified by noting that the apparent density of stars with and without the presence of dust goes as r/r0 = (do/d)3 = (10-0.2 A)3, so that an extinction of 1.5 magnitudes overestimates the distance by a factor of 2, and underestimates the density by a factor of 8. Because the extinction itself increases with distance, the density appears to drop uniformly away from the Sun. Today, great care is expended in determining extinctions when mapping out the spatial distributions of stars in the Milky Way and other nearby galaxies. Starcounts in Young Clusters: the Initial Mass Function Unlike the problematic situation encountered when fathoming stars in the Galaxy, uncertainties of extinction and distance virtually disappear when characterizing the stars in bound clusters. First, the clustered stars are all at nearly the same distance, and so are subject to similar amounts of foreground obscuration. By comparing a cluster’s extinction-corrected color-magnitude diagram with that containing stars of known distance and absolute magnitude (e.g. through "main-sequence fitting"), one can readily derive the cluster’s true distance and hence the absolute magnitudes of its constituent stars (see Endnote 3). The resulting distribution of absolute magnitudes (the stellar luminosity function) can then be converted into a distribution of stellar masses (see Endnote 11). The second key advantage of star clusters is that all of the stars in a cluster are of roughly identical age -- having formed from the same cloud. Therefore, the derived tally of stellar masses directly reflects the conditions that existed when the cluster stars first formed. For especially young clusters, even the highest-mass stars have yet to die off, and so the entire range of stellar masses is there for the counting. Astronomers refer to the stellar mass distribution at a cluster’s birth as the Initial Mass Function, or IMF. In clusters of the Milky Way, the IMF can be conveniently characterized as a power law that declines with increasing mass, such that N(M) dm = K M-a dM, where N(M) dM is the number of stars in the mass range M to M+dM, K is a constant, and the exponent for the so-called "local" or "Salpeter" IMF was found by Edwin Salpeter in 1955 to be a = 2.35 ± 0.3. A more common formulation for the IMF has the masses binned in logarithmic intervals. This logarithmic binning naturally arises from the observed luminosity function, where the stars are counted as a function of absolute magnitude MV -- the latter quantity being a logarithmically-compressed expression of luminosity (see Endnote 3). The conversion from absolute magnitude to mass preserves the logarithmic binning, so that the IMF becomes N(M) d(log M) = K MG d(log M), or log N(log M) d(log M) = log K + G log M d(log M), so that d[log N(M)]/d[log M] = G. Here, G is the so-called "slope" of the IMF which is related to the exponent a by G = 1 - a. Therefore, a Salpeter IMF with a = 2.35 corresponds to an IMF slope of G = -1.35. Such an IMF has the following breakdown of stars.
Table E5: The Local Stellar Initial Mass Function (IMF)
Similar proportions of high and low-mass stars are evident in the clusters that power the giant H II regions in M33 and the Magellanic Clouds, despite the widely varying environmental circumstances that exist in these galaxies. Moreover, the highest stellar masses that are observed in a particular cluster appear to depend mostly on the size and age of the cluster. As long as the cluster is large and young enough to populate the high-mass end, it will contain massive stars in similar proportions that are observed in other clusters. Somehow, the birth of low and high-mass stars in clouds entails self-regulating processes that do not critically depend on the clouds’ elemental makeup, dynamical circumstances, or other environmental factors. Some recent investigations have uncovered a possible enhancement of high-mass stars in chemically-enriched environments, but these results remain tentative. Once determined, the stellar IMF can be used as a "weighting function" to determine which types of stars dominate a cluster’s total mass and luminosity. For a young cluster that is fully populated with stars from a lower mass limit of 0.1 M[Suns] to an upper mass limit of 100 M[Suns], a Salpeter-type IMF will yield a mean mass of 0.3 M[Suns], corresponding to an M3-type star. The mean luminosity per star is much higher, however, amounting to 10,000 L[Suns], or the equivalent of a B3-type star with a mass of 16 Suns. This calculation underscores the widely differing stellar populations that respectively dominate the mass and light in a large, young cluster. The corresponding mass-to-light ratio is M/L @ 3 x 10-5, where M/L = 1 for the Sun. In the disk of our Galaxy, the M/L ratio is closer to that of the Sun, thus indicating the disk’s much greater age. Over some 12 billion years, successive generations of clusters and associations have formed and dissolved. The higher-mass stars quickly died off, while the intermediate and low-mass stars have accumulated over the eons. The resulting luminosity function is even more skewed towards the dim, low-mass stars than that evident in young clusters (see Figure 3.5). 20. Radiative Powering of H II Regions Ionized hydrogen (H II) regions are the luminous consequences of newborn hot stars fluorescing the natal gas that still surrounds them. The powering of the observed nebulosity is an uncertain mix of photo-ionization by the intense ultraviolet radiation from the hot stars along with shock heating and bulk flows by the stellar winds. In giant H II regions, supernova explosions can add as much mechanical heating as the lifetime-integrated stellar winds, further exciting the H II region. Here, we will concentrate on the radiative powering by the hot stars. The simplest situation is that of a single hot star surrounded by gas of uniform density and infinite extent. The star’s effective temperature and surface area determines its luminosity of H-ionizing photons, where each photon has an energy exceeding 13.6 electron-Volts and a corresponding wavelength less than 91.2 nanometers. For example, a main-sequence O6V-type star has an ionizing luminosity of about 1049 photons/s. The gas surrounding such a star will be mostly ionized, and will be heated to temperatures of 5,000 -- 10,000 Kelvins. When an H II region is in the state of ionization equilibrium, the ionizing luminosity of the star(s) equals the ionization rate in the gas, while the gas ionization rate equals the recombination rate. This balance can be expressed as Ni = aB nH ne Vol, where Ni is the total photo-ionizing luminosity, aB is the recombination rate per hydrogen atom, nH is the number density of hydrogen nuclei (protons), ne is the number density of available electrons, and Vol is the volume. Both densities are included in the recombination side of the equation, because it takes both a hydrogen nucleus and an electron to make a recombined atom -- in other words, it takes two to tangle. When assuming a single ionizing star and gas extending uniformly away to infinity, the most sensible volume to consider is that of a sphere. The above equation then becomes Ni = 4 p aB nH ne R3/3, which can be further simplified by letting nH = ne. This is valid only if the next most abundant element, helium, is not ionized. However, a hot O-type star will have a significant number of photons energetic enough to ionize helium at least once. Therefore, a more reasonable approximation is to let ne = nH + nHe = 1.1 nH, so that Ni = (1.1) 4 p aB nH2 R3/3. We can now solve this equation in terms of the H II region’s radius, within which the gas is in an ideal state of ionization equilibrium, getting RS = {3 Ni/(4 p aB nH2)}1/3, where RS refers to the radius of the so-called Stromgren sphere. Plugging in numbers, an O6V-type star like that powering the Orion Nebula has Ni = 1049 photons/s; the recombination rate to all hydrogen energy levels but the lowest (which would result in another ionizing photon and so doesn’t count) is measured in the laboratory to be aB = 10-13 cm3/s at temperatures typical of H II regions; and the hydrogen density in H II regions has typical values of nH = 100 cm-3. Such an H II region would have a Stromgren radius of about 14 light-years -- roughly twice the radial extent of the Orion Nebula. The main reason why the Orion Nebula does not extend as far as its predicted Stromgren radius is because it contains dust, whose absorption of ionizing photons reduces the total ionization rate in the gas. As the electrons recombine with the hydrogen nuclei, they quickly cascade down the available energy levels -- producing the characteristic spectrum of hydrogen line emission. Therefore, the nebula is seen to fluoresce in the presence of the hot star’s intense ultraviolet radiation field. By measuring the luminosity of Balmer Ha or Hb emission, one can calculate the total recombination rate in the gas, equate that to the total ionization rate, and so infer the total ionizing luminosity of the star(s) powering the H II region. This sort of "calorimetry" is often used to infer the hot stellar content of starbursting galaxies, when the individual stars cannot be resolved. The trigonometric relations linking observed radial velocities to rotational velocities and kinematic distances in the Galaxy are extensively covered in Galaxies and Galactic Structure by Debra Meloy Elmegreen (Prentice Hall, 1998), Galactic Astronomy by James Binney and Michael Merrifield (Princeton University Press, 1999), and Introductory Astronomy & Astrophysics by Michael Zeilik and Stephen Gregory (Saunders College Publishing, 1998). Considerable insights can be readily gained, however, by working in the opposite direction -- by first assuming a rotation curve, and seeing what happens. Let’s assume that the rotation velocity is constant with Galactocentric radius. Although such a "flat rotation curve" does not perfectly describe our Galaxy’s rotation, it provides a handy approximation to the rotation in the Milky Way and many other spiral galaxies. The rotation law is then simply v(R) = vo, and the angular velocity becomes w(R) = v(R)/R = vo/R. The differential rotation follows an even steeper dependence on radius, with dw(R)/dR = {dv(R)/dR (1/R)} + {v(R) (-1/R2)}, which at a constant rotation velocity of vo reduces to dw(R)/dR = -vo/R2. The shear flow in the disk can be approximated by Dv/DR @ (dv/dw) (dw/dR) = R (dw/dR) = -vo/R, so that a galaxy with vo = 200 km/s and R = 20,000 light-years has a shear flow of -10 km/s per 1,000 light-years of radial distance. By comparison, the measured shear flow in the Milky Way near the Sun is -8.9 km/s per 1,000 light-years -- fairly close to our approximation. The inner parts of the Milky Way and many other spiral galaxies have rotational velocities that rise quasi-linearly with radius for a few thousand light-years before flattening out. This behavior can be modeled as v(R) = C R, where the constant C corresponds to the slope of the rising rotation curve. The angular velocity is then w(R) = v(R)/R = C, resulting in zero differential rotation and shear flow. The inner galaxy rotates as a solid body -- like a compact disk. Where the rising rotation curve transitions into a flat rotation curve, the differential rotation and shear flow are highest. Often, important morphological features, such as bar ends and inner rings, are associated with this transitional region. In the Milky Way, the transition occurs within 2,000 light-years of the nucleus -- coincident with the terminus of a rapidly rotating inner disk of atomic and molecular gas. 22. The Nearest Galaxies -- the Local Group The following table summarizes the basic properties of the nearest galaxies. As of this writing, the Local Group of galaxies consists of about 40 members. More members are likely, however, as increasingly more sensitive surveys probe the sky for nearby faint dwarfs. Therefore, this table should not be regarded as a complete census of our galaxian neighborhood, especially at the faint end. Dynamical masses have been calculated for most of the Local Group systems. These estimates are based on either the gas rotation velocities or the stellar velocity dispersions in the galaxies (see Endnote 16). They range from close to a trillion (1012) solar masses for M31 and the Milky Way, to less than a few million Suns for the faintest dwarf ellipticals. The corresponding mass-to-light ratios are greatest for the faintest dwarfs, indicating that they harbor the highest proportions of dark matter.
Table E6: Selected Galaxies of the Local Group (in order of decreasing luminosity [increasing absolute magnitude, M(B)])
Notes to table: (1) Name of galaxy. (2) Galaxy classification, based mostly on the Hubble-Sandage system. Dwarf ellipticals are designated "dE," while dwarf irregulars are designated dIm. The dwarf spheroidal classification preferred by many investigators of the Local Group is designated "dSph." Peculiar morphologies are flagged with a "p." (3) Right Ascension (celestial longitude) in units of hours (hrs) and minutes (min), precessed to the 2000.0 epoch. (4) Declination (celestial latitude) in units of degrees (°) and arcminutes (¢ ), precessed to the 2000.0 epoch. (5) Distance from the Sun in units of light-years, determined mostly from observations of Cepheid and RR Lyrae variable stars. (6) Apparent magnitude at B-band (440 nm wavelength). (7) Absolute magnitude at B-band, based on apparent magnitude m(B) and the distance (see Technical Notes 2 and 3). References for table: Mateo, M. 1998, "Dwarf Galaxies of the Local Group," Annual Review of Astronomy and Astrophysics, Vol. 36, p. 435 Sandage, A. and Tammann, G. A. 1981, A Revised Shapley-Ames Catalog of Bright Galaxies, Carnegie Institution of Washington Publication 635, Washington, D.C. In the following table, 30 giant spiral (S), lenticular (S0), and elliptical galaxies (E) in the Local Group and other neighboring groups are listed. These galaxies have been selected for their apparent brightness in the sky (with m(B) < 10 mags) and high absolute luminosity (with M(B) < -19 mags and corresponding B-band luminosity exceeding 3 billion Suns [see Endnote 3]).
Table E7: Selected Giant Galaxies (in order of increasing distance from the Sun)
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