Technical Notes

LISTING OF Technical Notes

I. A GALAXY PRIMER

2. Numbers

12. Star Groups

II. NEARBY GALAXIES

25. Tidal Action

III. OUR GALAXIAN UNIVERSE

EPILOGUE

1. Distances

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.

2. Numbers

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

 Word Decimal Scientific one 1 100 two 2 2 x 100 ten 10 101 twenty 20 2 x 101> hundred 100 102 thousand 1,000 103 million 1,000,000 106 billion 1,000,000,000 109 trillion 1,000,000,000,000 1012

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.

3. Magnitudes

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.

4. Lookback Times

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.

5. Catalogues and Names

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.

7. Thermal Models of Galaxies

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).

8. Differential Rotation

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

 Johnson Filter Central Wavelength Bandwidth (FWHM) U (far-violet) 365 nm (3650 Å) 70 nm (700 Å) B (blue) 440 nm (4400 Å) 100 nm (1000 Å) V (yellow) ("visual") 550 nm (5500 Å) 90 nm (900 Å) R (red) 700 nm (7000 Å) 220 nm (2200 Å) I (far-red) 880 nm (8800 Å) 240 nm (2400 Å) J (near-infrared) 1250 nm (1.25 mm) 380 nm (0.38 mm) K (near-infrared) 2200 nm (2.2 mm) 480 nm (0.48 mm) L (near-infrared) 3400 nm (3.4 mm) 700 nm (0.70 mm) M (mid-infrared) 5000 nm (5.0 mm) 1200 nm (1.2 mm) N (mid-infrared) 10,400 nm (10.4 m m) 5700 nm (5.7 mm)

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.

11. Main Sequence Stars

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

 Type (1) (B - V) (mags) (2) T(eff) (oK) (3) MV (mags) (4) Lbol (Lbol[Sun]) (5) M (M[Sun]) (6) t (years) (7) O5 -0.32 50,000 -6.0 1.1 x 106 40 6.3 x 106 B0 -0.30 27,000 -4.1 6.5 x 104 17 1.2 x 107 B5 -0.16 16,000 -1.1 8.0 x 102 7 4.7 x 107 A0 0.00 10,400 +0.6 6.7 x 101 3.6 2.0 x 108 A5 +0.15 8,200 +2.1 1.3 x 101 2.2 7.7 x 108 F0 +0.30 7,200 +2.6 7.9 1.8 1.4 x 109 F5 +0.45 6,700 +3.4 3.7 1.4 3.1 x 109 G0 +0.60 6,000 +4.4 1.5 1.1 7.1 x 109 G5 +0.65 5,500 +5.2 7.4 x 10-1 0.9 1.5 x 1010 K0 +0.81 5,100 +5.9 4.2 x 10-1 0.8 2.3 x 1010 K5 +1.18 4,300 +8.0 9.8 x 10-2 0.7 3.8 x 1010 M0 +1.39 3,700 +9.2 5.1 x 10-2 0.5 1.5 x 1011 M5 +1.69 3,000 +12.3 6.7 x 10-3 0.2 1.0 x 1013

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.

12. Star Groups

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

 Phase Temperature T (oK) Number Density n (cm-3) Pressure/k P/k = nT (cm-3K) Cold Molecular 101 103 -- 106 104 -- 107 Cool Atomic 102 102 104 Warm Atomic 103 101 104 Warm Ionized (H II Regions) (Diffuse H II) 104 104 102 100 106 104 Hot Ionized (SNRs) (Coronal) 106 106 101 10-2 107 104

14. Doppler Shifts

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).

19. Counting Stars

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)

 Mass Range (M/M[Suns]) Logarithmic Mass Interval log (M/M[Suns]) Relative Number 30 -- 100 1.5 -- 2.0 1 10 -- 30 1.0 -- 1.5 5 3 -- 10 0.5 -- 1.0 22 1 -- 3 0.0 -- 0.5 114 0.3 -- 1.0 (-0.5) -- 0.0 501 0.1 -- 0.3 (-1.0) -- (-0.5) 2546

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.

21. Rotation in the Milky Way

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)])

 Name (1) Type (2) RA (hr:min) (3) Dec (°: ¢ ) (4) Distance (light-yrs) (5) m(B) (mags) (6) M(B) (mags) (7) M31 SbI-II 00:42.73 +41:16.1 2.5 x 106 4.4 -21.6 Galaxy SBbcI-II? 17:45.67 -20:00.5 2.7 x 104 -20.5? M33 ScII-III 01:33.85 +30:39.6 2.7 x 106 6.3 -19.1 LMC SBmIII 05:23.57 -69:45.4 1.6 x 105 0.6 -18.4 SMC ImIV-V 00:52.73 -72:49.7 1.9 x 105 2.8 -17.0 NGC 205 S0/E5p 00:40.37 +41:41.4 2.7 x 106 8.7 -16.0 M32 E2 00:42.70 +40:51.9 2.6 x 106 9.1 -15.8 NGC 3109 SmIV 10:03.12 -26:09.5 4.1 x 106 10.4 -15.2 IC10 Im 00:20.42 +59:17.5 2.7 x 106 12.9 -15.2 NGC 6822 ImIV-V 19:44.93 -14:48.1 1.6 x 106 9.8 -14.8 NGC 147 dE5/dSph 00:33.20 +48:30.5 2.4 x 106 10.3 -14.8 NGC 185 dE3p/dSph 00:38.97 +48:20.2 2.0 x 106 10.0 -14.7 IC 5152 SdmIV-V 22:02.70 -51:17.7 5.2 x 106 11.5 -14.5 IC 1613 ImV 01:04.90 +02:08.0 2.3 x 106 10.2 -14.2 Sextans A ImV 10:11.10 -04:42.5 4.7 x 106 11.7 -14.2 WLM ImIV-V 00:01.97 -15:27.8 3.0 x 106 11.0 -13.9 Sagittarius dIm/dSph 18:55.05 -30:28.7 7.8 x 104 4.1 -12.8 Fornax dE3/dSph 02:39.98 -34:27.0 4.5 x 105 8.2 -12.6 Pegasus dIm/dSph 23:28.57 +14:44.8 3.1 x 106 12.6 -12.3 SagDIG dIm 19:29.98 -17:40.7 3.5 x 106 13.9 -12.1 Leo A dIm 09:59.40 +30:44.7 2.2 x 106 12.9 -11.3 GR8 dIm 12:58.67 +14:13.0 5.2 x 106 14.8 -11.2 And I dE3/dSph 00:45.72 +38:00.4 2.6 x 106 13.5 -11.2 Leo I dE3/dSph 10:08.45 +12:18.5 8.1 x 105 10.9 -11.1 And II dE2/dSph 01:16.45 +33:25.7 1.7 x 106 13.3 -10.5 Sculptor dE3/dSph 01:00.15 -33:42.5 2.6 x 105 8.5 -10.4 LGS 3 dIm/dSph 01:03.88 +21:53.1 2.6 x 106 15.0 -9.9 DDO 210 dIm/dSph 20:46.77 -12:51.0 2.6 x 106 14.9 -9.9 And III dE5/dSph 00:35.28 +36:30.5 2.5 x 106 14.8 -9.7 Leo II dE0/dSph 11:13.48 +22:09.2 6.7 x 105 12.6 -9.0 Tucana dE/dSph 22:41.83 -64:25.2 2.9 x 106 15.7 -8.9 Carina dE4/dSph 06:41.62 -50:58.0 3.3 x 105 11.5 -8.6 Draco dE3/dSph 17:20.32 +57:54.8 2.7 x 105 11.8 -7.8 UrsaMinor dE5/dSph 15:09.18 +67:12.9 2.1 x 105 11.6 -7.6

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.

23. Nearby Giant Galaxies

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)

 Name(s) (1) Type Comment (2) RA (hr:min) Dec (o:¢ ) (3) Size (¢ ) m(B) (mags) (4) vel (km/s) Dist. (Mpc) (5) A(B) (mags) M(B) (mags) (6) Galaxy SBbc(rs)I-II "MilkyWay" 17:45.67 -20:00.5 0.0 0.008 -20.5? NGC 224 (M31) SbI-II "Andromeda 00:42.73 +41:16.1 190 x 60 4.4 -10 0.77* 1.7 -21.7 NGC 598 (M33) Sc(s)II-III "Pinwheel" 01:33.85 +30:39.6 70.8 x 41.7 6.3 670 0.85* 0.6 -19.0 NGC 55 Sc (edge-on) 00:15.13 -39:13.2 32.4 x 5.6 8.2 115 2.1 0.9 -19.3 NGC 3031 (M81) Sb(r)I-II 09:55.57 +69:04.1 26.9 x 14.1 7.9 124 3.6* 0.8 -20.8 NGC 2403 Sc(s)III 07:36.90 +65:35.9 21.9 x 12.3 8.9 2999 3.2* 0.6 -19.2 NGC 253 Sc(s) (starburst) 00:47.60 -25:17.4 27.5 x 6.8 8.1 504 2.8 0.7 -19.8 NGC 4736 (M94) RSab(rs) 12:50.90 +41:07.17 11.2 x 9.1 8.9 345 4.6 0.5 -19.9 NGC 6946 Sc(s)II 20:34.85 +60:09.4 11.5 x 9.8 9.7 336 4.5 1.5 -20.1 NGC 5236 (M83) SBc(s)II 13:37.00 -29:52.0 12.9 x 11.5 8.5 275 4.6 0.4 -20.2 NGC 5128 (Cen A) S0+S pec (AGN) 13:25.48 -43:01.0 25.7 x 20.0 7.9 251 4.6 1.27 -21.7 NGC 4826 (M64) Sab(s)II 12:56.75 +21:41.0 10.0 x 5.4 9.4 350 4.7 0.7 -19.7 NGC 4945 Sc (edge-on) 13:05.43 -49:28.0 20.0 x 3.8 9.6 275 4.7 1.3 -20.0 NGC 2903 Sc(s)I-II 09:32.17 +21:29.9 12.6 x 6.0 9.5 472 6.3 0.6 -20.1 NGC 4258 (M106) Sb(s)II (AGN) 12:18.95 +47:18.4 18.6 x 7.2 8.9 520 6.7 0.9 -21.2 NGC 5055 (M63) SbcII-III "Sunflower" 13:15.83 +42:01.7 12.6 x 7.2 9.3 550 7.3 0.5 -20.5 NGC 5194 (M51) Sbc(s)I-II "Whirlpool" 13:29.88 +47:11.9 11.2 x 6.9 8.6 541 7.3 0.4 -20.7 NGC 5457 (M101) Sc(s)I "Pinwheel" 14:03.22 +54:21.0 28.8 x 26.9 8.2 372 7.4* 0.3 -21.5 NGC 3627 (M66) Sb(s)II.2 "Black Eye" 11:20.25 +12:59.1 9.1 x 4.2 9.7 593 8.0 0.8 -20.6 NGC 4631 Sc (edge-on) 12:42.08 +32:32.4 15.5 x 2.7 9.8 606 8.0 0.9 -20.6 NGC 3521 Sb(s)II-III 11:05.82 -00:02.0 11.0 x 5.1 9.6 627 8.7 0.8 -20.8 NGC 6744 Sbc(r)II 19:09.77 -63:51.3 20.0 x 12.9 9.2 663 8.7 0.6 -21.1 NGC 1291 SBa 03:17.32 -41:06.5 9.8 x 8.1 9.4 738 10.0 0.2 -20.8 NGC 628 (M74) Sc(s)I 01:36.70 +15:47.2 10.5 x 9.5 9.8 861 11.5 0.3 -20.9 NGC 4594 (M104) Sa/Sb "Sombrero" 12:40:00 -11:37.4 8.7 x 3.5 9.3 873 11.3 0.9 -21.9 NGC 4472 (M49) E1/S0 12:29.78 +07:59.8 10.2 x 8.3 9.3 822 14.7 0.0 -21.5 NGC 4486 (M87) E0 "Virgo A" 12:30.83 +12:23.6 8.3 x 6.6 9.6 1136 14.7 0.0 -21.2 NGC 4649 (M60) S0 12:43.67 +11:33.1 7.4 x 6.0 9.8 1142 14.7 0.0 -21.0 NGC 1068 (M77) Sb(rs)II (AGN) 02:42.68 -00:00.9 7.1 x 6.0 9.5 1234 16.7 0.5 -22.0 NGC 1316 Sa pec "Fornax A" 03:22.70 -37:12.5 12.0 x 8.5 9.6 1713 22.3 0.3 -22.4

Notes to table:

(1) Name(s) of galaxy in the New General Catalogue (NGC) and Messier (M) listings.

(2) Galaxy classification, based mostly on the Hubble-Sandage system.

Proper names and other comments ... (AGN) refers to an active galactic nucleus.

(3) Right Ascension (celestial longitude) in units of hours (hrs) and minutes (min),

and Declination (celestial latitude) in units of degrees (o) and arcminutes (¢ ),

both coordinates precessed to the 2000.0 epoch.

(4) Angular dimensions along the major axis (2a) and minor axis (2b), in arcminutes (¢ ).

If the galaxy has a thin disk of circular shape, the respective dimensions provide a

measure of the galaxy’s inclination (i), such that {cos (i) @ b/a}.

Apparent magnitude at B-band, before correction for foreground extinction by dust.

(5) Radial velocity after correction for the peculiar motions of the Sun and Galaxy

with respect to the centroid of the Local Group.

Distance from the Sun in units of megaparsecs (Mpc), where 1 Mpc = 3.26 x 106

light-years. Distances with an asterix are based on observations of Cepheid variables

and subsequent application of the Period-Luminosity relation (see Chapters 7 & 12).

Distances of the remaining galaxies are estimated from the measured radial velocity

of each galaxy (or associated galaxy group) and application of the Hubble law, where

the assumed Hubble constant is Ho = 75 km/s/Mpc (see Chapters 12 & 15).

(6) B-band extinction due to foreground dust in our Galaxy and dust internal to the galaxy

itself (the latter extinction is based on the galaxy’s Hubble type and inclination).

Absolute magnitude at B-band, based on the extinction-corrected apparent magnitude

mo(B) and the distance (see Technical Notes 2 & 3).

References for table:

Madore, B. F. 2000, in Observers Handbook 2000, Ed. R. L. Bishop, Toronto, Canada:

Royal Astronomical Society of Canada, p. 264

Sakai, S., et al. 2000, "The Hubble Space Telescope Key Project on the Extragalactic

Distance Scale XXIV," Astrophysical Journal, Vol. 529, p. 678

Sandage, A. and Tammann, G. A. 1981, A Revised Shapley-Ames Catalog of Bright

Galaxies, Carnegie Institution of Washington Publication 635, Washington, D.C.

24. Starbursting and Interacting Galaxies

In the following table, 20 relatively nearby starbursting and interacting galaxies are listed. Their selection has no photometric basis (unlike in Table E7) but is based on their prominence in the scientific literature.

Table E8. Selected Starbursting and Interacting Galaxies

(in order of increasing distance)

 Name(s) (1) Type Comment (2) RA (hr:min) Dec (o:¢ ) (3) Size (¢ ) m(B) (mags) (4) Vel (km/s) Dist (Mpc) (5) A(B) (mags) M(B) (mags) (6) NGC 1569 (Arp 210) SmIV (starburst) 04:30.82 +64:50.9 3.6 x 1.8 11.9 144 2.2 3.0 -17.8 NGC 253 Sc(s) (starburst) 00:47.60 -25:17.4 27.5 x 6.8 8.1 504 2.8 0.7 -19.8 NGC 3034 (M82) (Arp 337) Amorphous (edge-on) (starburst) 09:55.87 +69:40.8 11.2 x 4.3 9.3 409 3.6 0.7 -19.2 NGC 4214 SBmIII (starburst) 12:15.66 +36:19.6 8.5 x 6.6 10.2 290 3.9 0.1 -17.8 NGC 5128 (Cen A) (Arp 153) S0+S pec (merger) (AGN) 13:25.46 -43:01.1 25.7 x 20.0 7.9 251 4.6 0.5 -20.9 NGC 5194/5 (M51) (Arp 85) Sc(s)+SB0 (interacting) "Whirlpool" 13:29.88 +47:11.9 11x7,6x5 9.0, 10.5 251 7.3 0.1 -20.5, -19.0 NGC 5457 (M101) (Arp 26) Sc(s)I (interacting) "Pinwheel" 14:03.22 +54:21.0 28.8 x 26.9 8.2 372 7.4 0.04 -21.2 NGC 2685 (Arp 336) S0 pec (polar ring) "Helix" 08:55.58 +58:44.0 4.5 x 2.3 11.9 1001 13.3 0.3 -19.0 NGC 3310 (Arp 217) Sbc(r) pec (starburst) 10:38.76 +53:30.2 3.1 x 2.4 11.2 1073 14.3 0.1 -19.7 NGC 1068 (M77) (Arp 37) Sb(rs)II (starburst) (HII/AGN) 02:42.68 -00:00.9 7.1 x 6.0 9.5 1234 16.7 0.1 -21.6 NGC 4038/9 (Arp 244) Sc+Sc pec (interacting) "Antennae" 12:01.88 -18:51.9 5x3, 3x2 11.2, 11.1 1391 18.5 0.2 -20.3, -20.4 NGC 520 (Arp 157) Amorphous (merger) 01:24.58 +03:47.5 2.2 x 1.1 12.1 2350 31.3 0.1 -20.5 NGC 7714 (Arp 284) (Mrk 538) SBb(s) pec (interacting) (HII/AGN) 23:36.23 +02:09.3 1.9 x 1.4 13.0 2994 39.9 0.2 -20.2 NGC 3690 (Arp 299) (Mrk 171) S+IBm pec (interacting) (HII/AGN) 11:28.50 +58:33.7 1x1, 2x1 11.8, 12.0 3096 41.3 0.1 -21.3, -21.1 NGC 7252 (Arp 226) "Atoms for RS0(r) (merger) Peace" 22:20.75 -24:40.7 1.9 x 1.6 12.9 4759 63.4 0.1 -21.2 Arp 220 S? (merger) (ULIRG) 15:34.95 +23:30.2 1.5 x 1.2 13.9 5473 73.0 0.22 -20.6 NGC 4676 (Arp 242) Im+SB0/a(s) (interacting) "Mice" 12:46.17 +30:43.6 2x1, 2x1 14.7, 14.4 6591 87.9 0.1 -20.1, -20.4 NGC 6240 Amorphous (merger) (ULIRG) 16:52.98 +02:24.0 2.1 x 1.1 13.8 7455 99.4 0.3 -21.4 Arp-Madore 0035-335 Ring (collision) "Cartwheel" 00:37.68 -33:43.0 1.1 x 0.9 15.2 9057 120.7 0.0 -20.2 Mrk 231 Sc(rs)? pec (HII/AGN) 12:56.24 +56:52.4 1.3 x 1.0 14.4 12715 169.5 0.0 -21.8

Notes to table:

(1) Name(s) of galaxy in the New General Catalogue (NGC), Messier (M), Arp, Arp-

Madore (AM), and Markarian (Mrk) listings.

(2) Galaxy classification, based mostly on the Hubble-Sandage system. These types are

most uncertain for the edge-on and strongly interacting systems.

Proper names and other comments ... (HII/AGN) refers to a mix of ionizing starburst

activity (HII) and non-stellar nuclear activity (AGN), while (ULIRG) denotes an ultra-

luminous infrared galaxy, where the infrared luminosity exceeds 1012 Suns.

(3) Right Ascension (celestial longitude) in units of hours (hrs) and minutes (min),

and Declination (celestial latitude) in units of degrees (o) and arcminutes (¢ ),

both coordinates precessed to the 2000.0 epoch.

(4) Angular dimensions along the major axis (2a) and minor axis (2b), in arcminutes (¢ ).

Apparent magnitude at B-band, before correction for foreground extinction by dust.

(5) Radial velocity after correction for the peculiar motions of the Sun and Galaxy with

respect to the centroid of the Local Group.

Distance from the Sun in units of megaparsecs (Mpc), where 1 Mpc = 3.26 x 106

light-years, based mostly on the measured radial velocity of the galaxy (or associated

galaxy group) and application of the Hubble law, where the assumed Hubble constant

is Ho = 75 km/s/Mpc (see Chapters 12 & 15).

(6) Extinction at B-band due to foreground dust in our Galaxy.

Absolute magnitude at B-band, based on the extinction-corrected apparent magnitude

mo(B) and the distance (see Technical Notes 2 & 3). This does not include the problematic correction for extinction internal to the galaxy, and hence underestimates the galaxy’s true B-band luminosity.

References for table:

Kennicutt, R. C., Schweizer, F., & Barnes, J. E. 1998, Galaxies: Interactions and Induced

Star Formation, Saas-Fee Advanced Course 26, Lecture Notes 1996, Swiss Society for

Astrophysics and Astronomy, Berlin, Germany: Springer-Verlag

Madore, B. F. 2000, in Observers Handbook 2000, Ed. R. L. Bishop, Toronto, Canada:

Royal Astronomical Society of Canada, p. 264

NASA/IPAC Extragalactic Database (NED) website, <http://nedwww.ipac.caltech.edu>

Sandage, A. and Tammann, G. A. 1981, A Revised Shapley-Ames Catalog of Bright

Galaxies, Carnegie Institution of Washington Publication 635, Washington, D.C.

25. Tidal Action

The dynamic effects of gravitational tides can be witnessed throughout the Universe -- from the oceanic tides on Earth, to the rings of icy debris surrounding Saturn, to the tidal bridges and tails of stars flung off interacting galaxies. All of these phenomena can be attributed to gradients of gravitational force across the respective bodies.

In the case of the Earth-Moon system, the Moon’s gravitation is most strongly felt on the side of Earth that is facing the Moon and most weakly felt on the opposite side. This gradient in the gravitational force across the Earth results in the oceans closest to the Moon accelerating towards the Moon by an amount that is greater than that of the Earth’s core. The oceans farthest from the Moon experience the least acceleration, and hence are seen to lag behind the Earth’s core. The net result are two tidally-induced oceanic bulges -- one directed towards the Moon, and the other bulging in the opposite direction. As the solid Earth spins underneath these two liquid bulges, each shoreline experiences two high tides per day. The "tails" and "bridges" of stars observed in interacting galaxies can be thought of as tidal bulges run riot.

Further insights on tidal action can be attained by considering how the gradient in gravitational force depends on the gravitating masses and the interacting geometry. From Newton’s "inverse square" law of universal gravitation, the gravitational force can be formulated as

FG = -GM1m/r2,

where M1 is the mass of the primary body, m is the mass of a test particle on the secondary body M2, r is the separation between mass centroids, G is Newton’s constant of universal gravitation, and the negative sign indicates the attractive nature of gravitational force. The gradient in the gravitational force can be derived by taking the spatial derivative of the above formulation, whereby

dFG/dr = +2GM1m/r3.

This tidal gradient varies as the inverse cube of the separation and is directed opposite to that of the gravitational force itself. If one considers the gradient across a particular body of radius R2, the corresponding tidal stress T from core to surface is

T = D F @ (dFG/dR) R2, or

T @ (2GM1m/r3) R2,

where the approximation applies only if the size of the secondary body is considerably smaller than the separation between bodies. Serious tidal disruption occurs at the so-called Roche limit, where and when a body’s tidal stress begins to exceed its own gravitational cohesion. Considering the tidal acceleration across such a body yields

aT = T/m @ (2GM1/r3) R2.

Setting the tidal acceleration equal to the body’s self gravity at its surface, such that

aT = aG = GM2/R22,

yields

r3/R23 @ 2M1/M2, or

r/R2 @ 1.3 (M1/M2)1/3.

A more careful analysis by Edouard Roche in 1850 that allowed for small separations yielded a somewhat higher coefficient of 2.44.

In the case of the Saturnian ring system, the Roche limit can be expressed in terms of the densities involved, such that

r @ 2.44 R2 (r1R13/r2R23)1/3, or

r @ 2.44 R1 (r1/r2)1/3,

where R1 is the radius of Saturn and r1/r2 is the ratio of densities in Saturn and the rings. Setting this ratio to unity yields a Roche limit that is about 2.5 times greater than the radius of Saturn -- neatly coinciding with the outer limits of the rings.

For bodies of equal mass and density, the critical separation is not much greater than the radius of the primary or secondary body. In other words, two galaxies of nearly equal mass must get very close to one another before their tidal effects can significantly disrupt one another. Such close and consequential interactions were prevalent in the very early Universe, when the density of matter was much greater. We can estimate when this occurred by considering the current density of giant galaxies in the local Universe. The average separation between giant galaxies is roughly 1 Mpc, or about 50 times greater than their individual radii. Reduction of this separation to the Roche limit, or roughly 2.5x the respective radii, occurred when the scale of the Universe was some 20 times smaller. Therefore, the last epoch of dominant tidal disruption should have ended at a redshift of z @ 20, and corresponding lookback time of roughly 13 Gyrs (see Endnote 28).

The foregone analysis ignores the glaring fact that galaxies are in motion. By considering the degree of motion (v), the volume density of galaxies (n), and the galaxy’s cross-sectional area (s), we can derive the collisional timescale (t) , whereby

t = 1/(nsv).

By setting the collisional timescale to a cosmologically interesting 1 Gyr and the rms velocity equal to that seen in galaxy groups (about 150 km/s), we can solve for the density of galaxies necessary to achieve this frequency of collisions, ie.

n = 1/(tsv).

For giant galaxies with cross-sectional areas corresponding to their Roche limits, the density amounts to roughly 5,000 galaxies per cubic Mpc. By contrast, groups in the local Universe have densities of no @ 1 giant galaxy/Mpc3. The higher density would have occurred when the relative scale of the Universe (R/Ro) was

R/Ro = (no/n)1/3,

or about 1/15 the size it is today -- again less than a billion or so years following the Big Bang. Since then, less disruptive interactions involving nearby dwarf galaxies have probably dominated the evolution of most giant galaxies.

These estimates notwithstanding, the actual rate of close interactions over cosmic time remains a hotly debated issue. Nearby examples of interacting galaxies, listed in Endnote 24 and discussed in Chapter 10, vividly demonstrate that close and tumultuous interactions have occurred up to the present day.

26. Galaxies with Active Nuclei

The nuclear activity in galaxies ranges from the mild LINERs in otherwise normal spiral galaxies, through the more powerful Seyfert and radio-lobe galaxies, to the most powerful quasars and blazars. The following 15 galaxies with active galactic nuclei (AGN) exemplify some of the variety found in the visible Universe ... out to a redshift of 5.5.

Table E9: Selected AGN Galaxies

(in order of increasing distance)

 Names(s) (1) Type Comment (2) RA (hr:min) Dec (o: ¢ ) (3) Size ( ¢ ) m(B) (mags) (4) Redshift t (Gyr) (5) NGC 3031 (M81) Sb(r)I-II (LINER) 09:55.57 +69:04.1 26.9 x 14.1 7.9 0.0004 0.012 NGC 5128 (Centaurus A) S0+S pec (radio-lobe) 13:25.48 -43:01.0 25.7 x 20.0 7.9 0.0008 0.015 NGC 4151 Sab(rs) (Seyfert 1) 12:10.54 +39:24.3 6.3 x 4.5 10-12 0.0033 0.045 NGC 4486 (M87/3C 274) (Virgo A) E0 (radio-lobe) (optical-jet) 12:30.83 +12:23.6 8.3 x 6.6 9.6 0.0038 0.053 NGC 1068 (M77/3C 71) Sab(rs)II (Seyfert 2) 02:42.68 -00:00.9 7x1 x 6.0 9.5 0.0041 0.054 NGC 4261 (3C 270) E3 (radio-lobe) 12:19.39 +05:49.5 4.1 x 3.6 11.4 0.0069 0.091 NGC 7714 (Mrk 538) SBb(s) pec (LINER) 23:36.23 +02:09.3 1.9 x 1.4 13.0 0.0100 0.13 NGC 1275 (3C 84) (Perseus A) E pec (Seyfert 2) 03:19.80 +41:30.7 2.2 x 1.7 11-13 0.0181 0.24 NGC 1265 (3C 83.1) E (radio-lobe) 03:18.26 +41:51.5 1.8 x 1.6 13.2 0.0258 0.34 Cygnus A (3C 405) Interacting (radio-lobe) 19:59.47 +40:44.0 <0.1 0.0569 0.74 BL Lacertae Blazar 22:02.72 +42:16.7 <0.1 14-17 0.0695 0.91 3C 273 Quasar (radio-lobe) (optical-jet) 12:29.11 +02:03.1 <0.1 12-13 0.1583 1.8 OJ 287 Blazar 08:54.81 +20:06.5 <0.1 12-16 0.3060 3.1 3C 48 Quasar 01:37.69 +33:09.6 <0.1 16-17 0.3670 3.5 RD J030117.. +002025 Quasar 03:01.28 +00:20.9 <0.1 24 5.50 11.1

Notes to table:

(1) Name(s) of galaxy in the New General Catalogue (NGC), Messier (M), Third

Cambridge Catalogue of Radio Sources (3C), and other listings.

(2) Galaxy classification, based mostly on the Hubble-Sandage system.

Proper names and other comments ... (LINER) refers to a low-ionization nuclear

emission region.

(3) Right Ascension (celestial longitude) in units of hours (hrs) and minutes (min),

and Declination (celestial latitude) in units of degrees (o) and arcminutes ( ¢ ), both

coordinates precessed to the 2000.0 epoch.

(4) Angular dimensions along the major axis (2a) and minor axis (2b), in arcminutes (¢ ).

Apparent magnitude at B-band, before correction for foreground extinction by dust.

Magnitude ranges are listed for highly variable galaxies.

(5) Cosmological Redshift (z), where z = Dl/l, after correction for Doppler shifts due to

motions of the Sun and Galaxy with respect to the centroid of the Local Group.

Lookback time (t) in giga-years (Gyrs) and corresponding distance (in light-Gyrs),

based mostly on the cosmological redshift (z) of the galaxy or associated galaxy group

and the assumption of free expansion, where t = to{1 - (1/[1 +z])}, and where the

Hubble time to = 1/Ho = 13.1 Gyrs, assuming Ho = 75 km/s/Mpc.

References for table:

NASA/IPAC Extragalactic Database (NED) website, <http://nedwww.ipac.caltech.edu>

Sandage, A. and Tammann, G. A. 1981, A Revised Shapley-Ames Catalog of Bright

Galaxies, Carnegie Institution of Washington Publication 635, Washington, D.C.

27. Galactic Black Holes

Like their stellar counterparts, galactic black holes are thought to occur whenever a concentration of gravitating matter imposes escape velocities that exceed light speed. Within the so-called event horizon of the black hole, no form of matter or radiation can escape the inexorable grip of the black hole’s gravitation. The fabric of space and arrow of time are warped beyond recognition within the event horizon, leaving us to theorize about the nature of this abyss.

The primary difference between galactic and stellar black holes is that the mass of a galactic black hole is so much greater. Instead of a few solar masses, galactic black holes weigh in at several million to several billion solar masses. Therefore, galactic black holes cannot be the collapsed cores of single massive stars, but must involve a tremendous concentration of stars -- or of their shredded remains.

Black Hole Basics:

One can quantify the masses and sizes of galactic black holes by equating the gravitational and kinetic energies, as would be the case at the brink of escape.

GMm/R = (mv2esc)/2,

where vesc is the escape velocity. Solving for the radius R yields

R = 2 GM/v2esc,

and setting the escape velocity equal to that of light (vesc = c) produces the relation

RS = 2 GM/c2,

where RS is the Schwarzchild radius (named after the German astrophysicist Karl Schwarzchild, who worked out this solution shortly after Albert Einstein published his general theory of relativity). This radius defines the size of the event horizon for a non-rotating spherical black hole. Modifications of this solution have been found for rotating black holes and for non-spherical shapes. Assuming the simple Schwarzchild solution, however, provides reasonable estimates for the sizes and densities of black holes of differing mass -- as itemized in the following table.

Table E10: Black Hole Dimensions

(In order of increasing black hole mass)

 Candidate Name(s) Type Mass (M/M[Sun]) Schwarzchild Radius Avg. Density (grams/cm3) Cygnus X-1 (HDE226868) Stellar BH 10.0 ± 5.0 30 km 1.0 x 10-4 ls 1.6 x 1014 LMC X-1 (0540-697) Stellar BH 7.0 ± 3.0 21 km 7.0 x 10-5 ls 3.7 x 1014 LMC X-3 (0538-641) Stellar BH 10.5 ± 3.5 31 km 1.0 x 10-4 ls 1.6 x 1014 M82 (NGC 3034) Amorphous Gal (Starburst Nuc) >500 >1.5 x 103 km 4.9 x 10-3 ls 7.3 x 1010 Milky Way (Sgr A*) SBbc(rs)I-II 2.6 x 106 7.7 x 106 km 26 ls 2.7 x 103 M32 (NGC 221) E2 3 x 106 8.9 x 106 km 30 ls 2.0 x 103 M77 (NGC 1068) Sb(rs)II (AGN) 1 x 107 3.0 x 107 km 99 ls = 1.6 lm 180 M31 (NGC 224) SbI-II 3 x 107 8.9 x 107 km 300 ls = 5 lm 20 M105 (NGC 3379) E0 1 x 108 3.0 x 108 km 16 lm 1.8 M84 (NGC 4374) E1 (radio jet) 1 x 109 3 x 109 km 160 lm = 2.7 lh 0.18 M87 (NGC 4486) E0 (radio-opt jet) 3 x 109 8.9 x 109 km 8.2 lh 2.0 x 10-3

References for table:

Cowley, A. P., 1992, "Evidence for Black Holes in Stellar Binary Systems," Annual

Review of Astronomy and Astrophysics, Vol. 30, p. 287

Kormendy, J. and Richstone, D. 1995, "Inward Bound: The Search for Supermassive Black Holes in Galaxy Nuclei," Annual Review of Astronomy and Astrophysics,

Vol. 33, p. 581

Notes to table:

With the exception of M82, the masses in the above table are determined from dynamical considerations of the observed velocities of stars and gas close to the center of mass (see Endnote 16). The mass of M82 is based on the period and amplitude of X-ray flickering.

The abbreviations "ls", "lm", and "lh" respectively refer to light-second, light-minute, and light-hour.

The above table underscores the tremendous differences between stellar and galactic black holes. As the masses of the respective black holes vault from several Suns to millions and billions of Suns, the Schwarzchild radii proportionately increase from several km to millions and billions of km. The corresponding light-crossing times increase from microseconds to hours, placing constraints on the flickering timescales of any emission produced by infalling material near the event horizons.

For stellar black holes, the average densities inside the event horizons are of nuclear magnitude. The galactic black holes are much roomier, hosting average densities ranging from that of the solar core (~100 grams/cm3), to that of water or the whole Sun (~1 grams/cm3), to that of air (~10-3 grams/cm3). The actual nature of the matter that is concentrated inside these netherworlds is beyond our empirical knowing.

The estimated masses of the galactic black holes trend with the masses of the host bulges -- the black holes amounting to 0.5 to 1.0 percent of the spheroidal components. This correlation indicates a generic relation between black holes and massive bulges. It also indicates that the quasar activity observed at high redshift -- if due to accreting galactic black holes -- was most likely the result of massive bulges having been built shortly after the Big Bang.

Black Hole Energetics

To get power from a black hole, the "central engine" must be fueled. Astronomers envision that galaxy mergers and other less drastic interactions between galaxies can send gas, stars, and stellar remnants plummeting into a galaxy’s nucleus and whatever black hole may lurk there. As matter hurtles toward the black hole’s event horizon, the conversion of gravitational potential energy into kinetic energy provides a vast supply of mechanical power. This can be formulated by considering the gravitational potential energy released when a test particle falls from infinite radius to somewhere near the Schwarzchild radius of the black hole

G.E. = GMm/RS,

where M is the mass of the black hole, and m is the test particle’s mass. The corresponding power is obtained by taking the time derivative such that

Power = d(G.E.)/dt = (GM/RS)(dm/dt),

where dm/dt is the mass inflow rate. For example, a billion solar-mass black hole hosting a mass inflow rate of 1 M[Sun]/yr could generate up to 7 trillion solar luminosities worth of mechanical power.

Once the material crashes onto the accretion disk, the kinetic energy is then thermalized and re-released in the form of X-rays and outflowing jets of magnetized plasma. The efficiency of this conversion determines the actual radiative and mechanical luminosities that are observed, e.g.

Lrad = (eff) Power,

where the radiative efficiency (eff) is thought to be somewhere between 1 and 10 percent. Given such high efficiencies, the most luminous quasar activity can be explained as arising from the accretion of matter near a supermassive black hole.

An accreting black hole’s radiative luminosity is ultimately limited by the radiation pressure which this same luminosity engenders on the black hole’s accreting environment. Once the radiation pressure exceeds the gravitational binding, the influx of "fuel" is squelched until the pressure is again below the limit. The equality between radiation pressure force and gravitational force can be formulated as

s Lrad/(4pR2)c = GMmH/R2,

where M is the mass of the black hole, mH is the mass of the dominant hydrogen nucleus, and the electron-photon cross section (s = 6.65 x 10-25 cm2) can be regarded as the relevant efficiency for the radiation pressure term. Solving for the luminosity yields

Lrad = 4pcGMme/s,

or

Lrad/L[Sun] = 3.3 x 104 M/M[Sun]

This threshold state is known as the Eddington Limit and is thought to apply to feeding AGNs, providing yet another method of "weighing" the putative black holes within.

For example, the highest luminosities observed in quasars (L @ 1013 L[Suns]) would require something like a billion solar-mass black hole emitting at the Eddington Limit.

The corresponding mass inflow rate would be something like 100 Suns/yr, assuming an energy conversion efficiency of 1%.

28. Galaxy Clusters and Superclusters

Our Local Group of galaxies is one of about 50 other groups within a radius of about 20 Mpc. Each group spans about 1 Mpc, contains ten or so galaxies brighter than the dwarf threshold of M(V) = -16 mags, and has a typical density of 10/Mpc3 -- or roughly 10 times the mean density of galaxies. Some of these groups are concentrated into larger clouds, such as the Coma-Sculptor cloud which spans about 10 Mpc and encompasses the Local, Sculptor, M81, M101, and Canes Venatici groups.

Richer concentrations of galaxies are known as clusters. Nearby examples include the Ursa Major and Virgo clusters. Galaxy clusters typically measure 2-10 Mpc across, contain 10-1000 bright galaxies and dark matter amounting to 1012-1015 solar masses. Typical mass-to-light ratios are in the hundreds, indicating a preponderance of dark matter.

The nearby groups and clusters are all part of the Virgo (or "Local") supercluster, a vast, elongated aggregation measuring 40 Mpc across and containing 1016 Suns worth of mass. Other superclusters include Coma-Leo, roughly 90 Mpc away, Hercules-A2199 at 150 Mpc, and Corona Borealis 290 Mpc away. More extensive aggregates delineate colossal filaments and shells, as exemplified by the Perseus-Pegasus supercluster filament (or "chain") which spans about 200 Mpc over distances of 70-160 Mpc, and the "Great Wall" which appears to connect the Coma and Hercules superclusters (spanning 117o on the sky, or about 150 Mpc at a distance of about 400 Mpc). On the largest observable scales, the overall distribution of galaxies suggests a cellular structure with each "cell" (and corresponding "void") measuring roughly 100 Mpc across.

Table E11: Selected Galaxy Clusters

(in order of increasing distance)

 Name(s) (1) R.A. (hr:min) Dec (o: ¢ ) (2) Vel (km/s) Dist (Mpc) (3) Richness Type (4) Membership (5) Local Group 0 0 0 III Coma-Sculptor Cld; Virgo SC M81 Group 09:55.6 +69:04 240 3.6* 0 III Coma-Sculptor Cld; Virgo SC Ursa Major 11:57.2 +49:17 957 20.7T 0 III Virgo SC Virgo 12:26.5 +12:43 1026 16.1* 2? III Virgo SC Fornax (Abell 373S) 03:38.5 -35:27 1486 19.0* 0 I Southern SC Hydra (Abell 1060) 10:36.9 -27:32 3454 60.2T 1 III Hydra-Centaurus SC Centaurus (Abell 3526) 12:48.9 -41:18 2756 44.7T 0 I-II Hydra-Centaurus SC Pegasus (Abell 2594??) 23:20.5 +08:11 4109 55.8T 1 II-III Norma (Abell 3627) 16:15.5 -60:54 4707? 62.8? 1 I Perseus (Abell 426) 03:18.6 +41:32 5490 73 2 II-III Perseus-Pegasus Chain Stephan’s Quintet (Arp 319) 22:36.0 +33:58 6446 85.9 Compact Group Coma (Abell 1656) 12:59.8 +27:58 6889 88.6T 2 II Coma/Leo SC; Great Wall Hercules (Abell 2151/2) 16:05.2 +17:43 11,100 148 2 III Hercules SC; Great Wall Corona Borealis (Abell 2065) 15:22.7 +27:43 21,600 288 2 III Corona Borealis SC

Notes to table:

(1) Name(s) of galaxy group or cluster. Abell # or A# refers to the listing in George

Abell’s Catalog of Galaxy Clusters.

(2) Right Ascenscion (celestial longitude) in units of hours (hrs) and minutes (min),

and Declination (celestial latitude) in units of degrees (o) and arcminutes (¢ ),

both coordinates precessed to the 2000.0 epoch.

(3) Radial velocity after correction for the peculiar motions of the Sun and Galaxy with

respect to the centroid of the Local Group.

Distance from the Sun in units of megaparsecs (Mpc), where 1 Mpc = 3.26 x 106

light-years. Distances with an asterix are based on observations of Cepheid variables

and other individual stars. Distances with a cross are based on observations of the

H I linewidths and far-red magnitudes of the member galaxies, and application of the

Tully-Fisher relation (see Chapter 12). The remaining distances are estimated from

the measured radial velocity and application of the Hubble law, where the assumed

Hubble constant is Ho = 75 km/s/Mpc.

(4) Richness class, where 0 ==> 30-49 galaxies within 2 mags of 3rd brightest member,

1 ==> 50-79 galaxies, 2 ==> 80-129 galaxies, 3 ==> 130-199 galaxies, etc.

Type, where I ==> regular with strong central concentration, and III ==> irregular

with weak or no central concentration.

(5) Membership of group or cluster to larger cloud (Cld), supercluster (SC), etc.

References for table:

Abell, G. O., et al. 1989 "A Catalog of Rich Clusters of Galaxies," Astrophysical Journal

Supplement, Vol. 70, p. 1 (see also <http://heasarc.gsfc.nasa.gov>)

NASA/IPAC Extragalactic Database (NED) website, <http://nedwww.ipac.caltech.edu>

Sakai, S., et al. 2000, "The Hubble Space Telescope Key Project on the Extragalactic

Distance Scale XXIV," Astrophysical Journal, Vol. 529, p. 678

Tully, R. B. 1987, Nearby Galaxies Atlas, Cambridge, UK: Cambridge University Press

Ibid 1988, Nearby Galaxies Catalog, Cambridge, UK: Cambridge University Press

Zeilik, M., and Gregory, S. A. 1998, Introductory Astronomy and Astrophysics, 4th Ed.,

Orlando, FL: Saunders College Publishing, Ch. 23

29. Cosmological Relations

The science of cosmology addresses the most far-reaching questions of the Universe -- including those of its origin, structure, dynamics, and fate. Although cutting-edge research in cosmology requires a thorough grounding in gravitational physics (ie. general relativity), much can be gained by simply considering the overall scale of the Universe and how this evolving scale relates to basic observables such as the redshift, and derived properties such as the expansion rate, lookback time, temperature, and density.

The scale of the Universe (R) refers to the spacing of galaxies over time. By comparing this scale with that at the present day (Ro), one need not refer to the spacing of any particular set of galaxies. This is simply done by taking the ratio R(t)/Ro, where Ro is equal to R(t) at the current epoch (t = to).

Redshifts

In an expanding Universe, the scale increases with time. Any propagating waves of light stretch along with the expanding space. The resulting wavelengths are simply related to the scale by

lo/le = Ro/Re,

where lo and le respectively refer to the observed and emitted wavelengths. This can be re-written as

(le + Dl)/le = Ro/Re,

or, in terms of the observed redshift (z), as

(1 + z) = Ro/Re,

where z = Dl/le. Conversely, the scale of the Universe -- as traced by galaxies at a given redshift -- can be gauged in terms of that redshift as

Re/Ro = 1/(1 + z),

The corresponding density, being inversely proportional to the cube of the scale factor, is

ne/no = (1 + z)3.

For example, the epoch of greatest quasar activity is observed at redshifts of z @ 3. The scale of the Universe was 4 times smaller then, with a density that was 64 times greater.

Temperatures

In a thermalized Universe, the temperature solely determines the spectral energy distribution (see Endnote 10). The corresponding wavelength of peak intensity is found to be inversely proportional to the temperature,

l(peak) = C/T,

where C = 0.29 cm-K. As the wavelength increases with the expanding scale of the Universe, the temperature declines commensurately, ie.

To/Te = le/lo = 1/(1 + z).

An important application of this relation pertains to the cosmic microwave background radiation (CMBR) and what it signifies. The dominant thinking is that the CMBR originated from the epoch when the expanding and cooling Universe was changing phase from an ionized plasma to a neutral atomic gas which would then be transparent to its own radiation. Studies of nearby H II regions, where similar conditions exist, suggest that the relevant temperature was about 3,000 K. As observed, the CMBR has the spectral energy distribution of a 2.7 K black body. Solving the above relation for the redshift yields

z = Te/To - 1,

which at the epoch of decoupling would yield a redshift of z @ 1100. Therefore, the Universe was 1100 times more compact and 1.4 x 109 times denser during this critical epoch.

Energies

Prior to the epoch of decoupling, the energy density of the Universe was dominated by radiation. Each photon has an energy that is inversely proportional to its wavelength, ie.

Eph = hc/l.

As the Universe expanded, the wavelength increased with the scale factor, and so the photon energy decreased as 1/R. Meanwhile, the number density of photons (nph) decreased as 1/R3. The resulting energy density (rph) during the radiation epoch declined as the product of the number density and energy per photon, ie.

rph = nph Eph µ 1/R4.

By contrast, the energy density of matter (rm) declined simply as 1/R3, so that the ratio of energy densities was

rph/rm µ 1/R.

Shortly before the epoch of decoupling, the energy density of matter began to overtake the photon energy density. Today, the ratio of CMBR photons to baryons is still about a billion -- having been set in the very early Universe. However, the ratio of energy densities has declined from near unity at decoupling to something like rph/rm @ 1/1100.

Free Expansion

Translating redshift into lookback time depends critically on the history of expansion. For example, an accelerating expansion will yield greater lookback times at a given redshift than a constant-velocity or decelerating expansion. We can best begin to understand these complexities by first considering the simple case of free expansion at constant velocity. Here, there is no gravitating mass to slow things down, or the decelerating matter is perfectly balanced by some sort of accelerating dark energy. The upshot is that the scale of the Universe increases linearly with cosmic time,

R(t)/Ro = t/to = 1/(1 + z),

so that the time (t) is simply

t = to/(1 + z),

where (to) is the age of the current epoch. The lookback time (t) is

t = to - t = to{1 - (1/[1 + z])}.

For example, the lookback time of a z = 5 quasar is 83% of the total age of the Universe, or about 12.5 Gyr in a Universe that is 15 Gyr old.

In the linearly expanding scenario, the expansion rate is simply

dR(t)/dt = Ro/to.

Dividing this rate by the scale at time (t) yields the so-called Hubble constant (H), where

H = [dR(t)/dt]/R(t) = 1/t.

In the case of free expansion, the Hubble "constant" is seen to actually decrease with time. At the current epoch, Ho = 1/to, so that the age of the Universe (to) is simply

to = 1/Ho.

The current-epoch Hubble constant (Ho) can be derived by measuring the distances and redshifts of many galaxies. Each distance (d) provides a measure of the scale factor (Ro), while each redshift yields the rate of expansion ([dR/dt]o @ cz) at that scale, the result being

Ho = [dR/dt]o/Ro @ cz/d,

or the more frequently used version of the Hubble Law

Ho = Vr/d,

where (Vr) is usually expressed in units of km/s, and (d) is in units of Mpc. In these units the Hubble time becomes

to = 9.8 (100/Ho) Gyrs,

or 13--15 Gyrs for Ho = 75--65 km/s/Mpc.

In the following table, the relation between redshift and lookback time in a freely expanding Universe is delineated for a variety of Hubble constants.

Table E12: Lookback Times

(assuming free expansion)

 Redshift (z = Dl/l) t/to t(Gyrs) (Ho = 50) t(Gyrs) (Ho = 75) t(Gyrs) (Ho = 100) 0.01 0.01 0.19 0.13 0.10 0.03 0.03 0.57 0.38 0.28 0.10 0.09 1.79 1.19 0.89 0.30 0.23 4.55 3.02 2.26 1.0 0.50 9.85 6.55 4.90 3.0 0.75 14.8 9.82 7.35 10.0 0.91 17.9 11.9 8.91 30.0 0.97 19.1 12.7 9.48 100 0.99 19.5 13.0 9.70 ¥ 1.0 19.7 13.1 9.80

Decelerating and Accelerating Expansion

If the expansion rate changes over time, then a dimensionless deceleration (or acceleration) parameter (qo) can be defined

qo = {-R(d2R/dt2)/(dR/dt)2}o,

such that values greater than 0 indicate deceleration, while negative values connote accelerating expansion. The special value of (qo = 1/2) indicates a Universe whose decelerating expansion would ultimately come to a halt. Were our Universe on such a critical trajectory, then the total age of the Universe would equal 2/3 the Hubble time, or only 9 Gyrs for Ho = 75 km/s/Mpc. Clearly such a short age would violate the ages found for globular clusters and other "clocks" in the Universe. More likely, our Universe is on a trajectory of nearly free expansion -- and quite possibly an accelerating expansion.

One simple form of accelerating expansion can be explored by letting the Hubble "constant" be constant for all time. In that case,

H = (dR/dt)/R = Ho,

so that

dR/R = Ho dt.

Integrating this relation from to to t yields

ln [R(t)/Ro(to)] = Ho (t - to) = (t/to - 1),

or

R(t) = Ro exp (t/to - 1),

where the Hubble time (to) in this case equals the e-folding expansion time. This particular form of exponential expansion leads to improbably long lookback times at redshifts exceeding 5 and so has limited relevance over the majority of cosmic history. Such exponential growth may have prevailed, however, during the putative inflationary epoch -- though on a vastly shorter timescale -- and may now be taking hold in milder form once again.

30. The Drake Equation -- A Prospectus for Intelligent Life

The emerging field of astrobiology got its first major boost in 1961, when ten radio technicians, astronomers, and biologists convened in Green Bank, West Virginia, to address the prospects for intelligent, communicative life beyond Earth. In preparation for the meeting, Frank Drake of the National Radio Astronomy Observatory (NRAO) developed an equation that formulates the sundry (im)probabilities that combine to yield an estimate for the number of telecommunicating planets in the Milky Way. Beginning with the total number of stars in our Galaxy, the eponymous "Drake Equation" lays down a gauntlet of challenging questions -- some of which are still as uncertain as ever.

The equation itself has a disarming simplicity, consisting of just 7 factors in its most succinct form:

N(comm) = N(star) x f(sun) x f(plan) x n(hab) x f(life) x f(int) x f(comm),

where:

N(comm) = number of intelligent, communicative civilizations in the Milky Way,

N(star) = number of stars in the Milky Way,

f(sun) = fraction of stars similar to the Sun,

f(plan) = fraction of sunlike stars with planets,

n(hab) = number of planets occupying habitable zone of sunlike star,

f(life) = fraction of habitable planets with life,

f(int) = fraction of inhabited planets with intelligent beings

f(comm) = fraction of planets with intelligent species that are technologically communicative.

To see how these co-factors collectively winnow the prospects, let’s consider each factor in its turn, and propagate the results.

N(star) = number of stars in the Milky Way (~100 x 109):

Although an actual count of all the stars in the Milky Way is an impossibility, we can obtain a near-complete census of the stars within 5 parsecs of the Sun. The resulting density of 0.1 stars/pc3, when combined with models of the overall distribution of stars in the Milky Way, yields a total number of about 100 - 300 billion stars. These results are consistent with observations of the total starlight from other nearby giant spiral galaxies.

f(sun) = fraction of stars similar to the Sun (~0.1):

To enable the evolution of life, a stable long-lasting star is preferred. Stars significantly more massive than the Sun are too short-lived, while stars of much lower mass are prone to violent flaring. For these reasons, stars of spectral type F, G, and K make the most promising candidates (see Endnote 11). The census of stars in the Solar Neighborhood suggests a fraction of 0.2, or more conservatively, 0.1. That leaves about 10 billion Sun-like stars.

f(plan) = fraction of Sun-like stars with planets (~0.1):

Two pieces of evidence indicate that planetary systems are fairly common. One is the fraction of stars in binary systems. Roughly 75% of all the stars are parts of binary or multiple star systems. These stellar bunchings tend to be dynamically hostile to any associated planetary systems. The remaining 25% of stars are thought to be more likely hosts of planets in stable orbits. The second piece of evidence is the actual harvest of planetary systems that has been obtained from tracking the Doppler-shifted light of nearby stars. Of the approximately 1,000 stars that have been surveyed to date, about 70 appear to be in the gravitating presence of planets. These dynamically-based surveys are biased towards the detection of massive planets which are very close to their host stars. However, they provide a reasonable lower limit of 10% for the fraction of stars with planets, leaving about 1 billion Sun-like stars.

n(hab) = number of habitable planets per Sun-like planetary system (~0.1):

Here, we begin to run into speculative territory. Current observing technology is insufficient to detect planets that are significantly less massive than Jupiter, ie. Earths. Therefore, we are left with extrapolations from our own situation. Of the nine planets in the Solar System, only Earth occupies the so-called "water-zone," the annular region whose distance from the Sun allows water to be in liquid state. Earth barely qualifies, the greenhouse effect of its atmosphere providing enough additional heating to keep the surface from freezing over -- most times. The subsurfaces of Mars and Europa may also harbor liquid water. The biochemistry of life on Earth relies on water as the primary chemical solvent. Other chemicals, such as ammonia, may qualify as solvents in other more frigid worlds. Lacking further insights, our "water zone" requirement suggests a number of 1 habitable planet per system, or more conservatively 1 habitable planet every 10 systems. We are then left with about 100 million habitable planets in the Milky Way.

f(life) = fraction of habitable planets where life arises (~0.1):

Without any hard evidence yet for life of any kind beyond Earth, astrobiologists are still very optimistic about the prospects for life evolving on habitable planets elsewhere in the Milky Way. The ubiquity of complex organic molecules in star-forming regions and of amino acids in meteorites tell us that the necessary ingredients are available. Moreover, the uncanny flourishing of microbial life in the most extreme terrestrial environments attests to the tenacious qualities of single-celled lifeforms. A naive consideration of the terrestrial record reveals a living history that goes back a billion years, or roughly 20% of the 4.6 billion-year age of Earth. If we shamelessly extrapolate from our own situation, then we might guess that 1 in 10 habitable planets actually sports life of some kind. Our educated guess would yield about 10 million life-bearing planets in the Milky Way.

f(int) = fraction of inhabited planets with intelligent beings (~10-3):

Once again, our lack of information beyond Earth leaves us grasping at our own reflection. Using relative lifetimes as our guide, we find that human-like intelligence has marked the Earth for only a few million years, or about 1/1,000 the history of life on Earth. Perhaps intelligent life forms will continue to prevail on Earth for eons to come -- thus upping the extrapolated odds. For now, however, a fraction of ~10-3 is as good a guess as any. Our tally of the Milky Way is now reduced to about 10 thousand planets with intelligent life.

f(comm) = fraction of planets with intelligent species that are technologically communicative (~10-4):

Here be dragons. This fraction critically depends on how long a technologically communicative species can functionally survive. To date, our species has been transmitting radio energy for about 102 years, or roughly 10-4 of our history as hominids. Perhaps we can keep this up for another thousand years or more -- who knows! Meanwhile our guessing game would lead to the following total number of telecommunicating planets in the Milky Way: N(comm) ~ 1. We may be it!

Far more detailed considerations of the amazing happenstances that presaged the emergence of intelligent life on Earth and of their implications regarding the prospects for intelligence elsewhere in the Galaxy can be found in Rare Earth, by P. D. Ward and D. Brownlee, 2000 (New York: Copernicus/ Springer-Verlag).