Chapter 2 Gfe PDF

2 Observational Facts Observational astronomy has developed at an extremely rapid pace. Until the end of the 1940s observational astronomy was limited to optical wavebands. Today we can observe the Universe at virtually all wavelengths covering the electromagnetic s...

2 Observational Facts Observational astronomy has developed at an extremely rapid pace. Until the end of the 1940s observational astronomy was limited to optical wavebands. Today we can observe the Universe at virtually all wavelengths covering the electromagnetic spectrum, either from the ground or from space. Together with the revolutionary growth in computer technology and with a dramatic increase in the number of professional astronomers, this has led to a flood of new data. Clearly it is impossible to provide a complete overview of all this information in a single chapter (or even in a single book). Here we focus on a number of selected topics relevant to our forthcoming discussion, and limit ourselves to a simple description of some of the available data. Discussion regarding the interpretation and/or implication of the data is postponed to Chapters 11–16, where we use the physical ingredients described in Chapters 3–10 to interpret the observational results presented here. After a brief introduction of observational techniques, we present an overview of some of the observational properties of stars, galaxies, clusters and groups, large scale structure, the intergalactic medium, and the cosmic microwave background. We end with a brief discussion of cosmological parameters and the matter/energy content of the Universe. 2.1 Astronomical Observations Almost all information we can obtain about an astronomical object is derived from the radiation we receive from it, or by the absorption it causes in the light of a background object. The radiation from a source may be characterized by its spectral energy distribution (SED), fλ dλ , which is the total energy of emitted photons with wavelengths in the range λ to λ + dλ. Technology is now available to detect electromagnetic radiation over an enormous energy range, from low frequency radio waves to high energy gamma rays. However, from the Earth’s surface our ability to detect celestial objects is seriously limited by the transparency of our atmosphere. Fig. 2.1 shows the optical depth for photon transmission through the Earth’s atmosphere as a function of photon wavelength, along with the wavelength ranges of some commonly used wavebands. Only a few relatively clear windows exist in the optical, near-infrared and radio bands. In other parts of the spectrum, in particular the far-infrared, ultraviolet, X-ray and gamma-ray regions, observations can only be carried out by satellites or balloon-borne detectors. Although only a very restricted range of frequencies penetrate our atmosphere, celestial objects actually emit over the full range accessible to our instruments. This is illustrated in Fig. 2.2, a schematic representation of the average brightness of the sky as a function of wavelength as seen from a vantage point well outside our own galaxy. With the very important exception of the cosmic microwave background (CMB), which dominates the overall photon energy content of the Universe, the dominant sources of radiation at all energies below the hard gamma-ray regime are related to galaxies, their evolution, their clustering and their nuclei. At radio, far- UV, X-ray and soft gamma-ray wavelengths the emission comes primarily from active galactic 25 26 Observational Facts Fig. 2.1. The altitude above sea level at which a typical photon is absorbed as a function of the photon’s wavelength. Only radio waves, optical light, the hardest γ -ray, and infrared radiation in a few wavelength windows can penetrate the atmosphere to reach sea level. Observations at all other wavebands have to be carried out above the atmosphere. nuclei. Galactic starlight dominates in the near-UV, optical and near-infrared, while dust emis- sion from star-forming galaxies is responsible for most of the far-infrared emission. The hot gas in galaxy clusters emits a significant but non-dominant fraction of the total X-ray back- ground and is the only major source of emission from scales larger than an individual galaxy. Such large structures can, however, be seen in absorption, for example in the light of distant quasars. 2.1.1 Fluxes and Magnitudes The image of an astronomical object reflects its surface brightness distribution. The surface brightness is defined as the photon energy received by a unit area at the observer per unit time from a unit solid angle in a specific direction. Thus if we denote the surface brightness by I, its units are [I] = erg s−1 cm−2 sr−1. If we integrate the surface brightness over the entire image, we obtain the flux of the object, f , which has units [ f ] = erg s−1 cm−2. Integrating the flux over a sphere centered on the object and with radius equal to the distance r from the object to the observer, we obtain the bolometric luminosity of the object: L = 4π r 2 f , (2.1) with [L] = erg s−1. For the Sun, L = 3.846 × 1033 erg s−1. The image size of an extended astronomical object is usually defined on the basis of its isopho- tal contours (curves of constant surface brightness), and the characteristic radius of an isophotal contour at some chosen surface brightness level is usually referred to as an isophotal radius of the object. A well-known example is the Holmberg radius defined as the length of the semimajor axis of the isophote corresponding to a surface brightness of 26.5 mag arcsec−2 in the B-band.Two other commonly used size measures in optical astronomy are the core radius, defined as the  2.1 Astronomical Observations 27 CMB FIB NIB XRB optical GRB radio IR UV X−ray Fig. 2.2. The energy density spectrum of cosmological background radiation as a function of wavelength. The value of νIν measures the radiation power per decade of wavelength. This makes it clear that the cosmic microwave background (CMB) contributes most to the overall background radiation, followed by the far- (FIB) and near-infrared (NIB) backgrounds, the X-ray background (XRB) and the gamma-ray background (GRB). [Courtesy of D. Scott; see Scott (2000)] radius where the surface brightness is half of the central surface brightness, and the half-light radius (also called the effective radius), defined as the characteristic radius that encloses half of the total observed flux. For an object at a distance r, its physical size, D, is related to its angular size, θ , by D = rθ. (2.2) Note, though, that relations (2.1) and (2.2) are only valid for relatively small distances. As we will see in Chapter 3, for objects at cosmological distances, r in Eqs. (2.1) and (2.2) has to be replaced by the luminosity distance and angular diameter distance, respectively. (a) Wavebands and Bandwidths Photometric observations are generally carried out in some chosen waveband. Thus, the observed flux from an object is related to its SED, fλ , by ! fX = fλ FX (λ )R(λ )T (λ ) dλ. (2.3) Here FX (λ ) is the transmission of the filter that defines the waveband (denoted by X), T (λ ) repre- sents the atmospheric transmission, and R(λ ) represents the efficiency with which the telescope plus instrument detects photons. In the following we will assume that fX has been corrected for atmospheric absorption and telescope efficiency (the correction is normally done by calibrating the data using standard objects with known fλ ). In this case, the observed flux depends only on the spectral energy distribution and the chosen filter. Astronomers have constructed a variety of 28 Observational Facts Table 2.1. Filter characteristics of the UBVRI photometric system. Band: U B V R I J H K L M λeff (nm): 365 445 551 658 806 1220 1630 2190 3450 4750 FWHM (nm): 66 94 88 138 149 213 307 390 472 460 M⊙ : 5.61 5.48 4.83 4.42 4.08 3.64 3.32 3.28 3.25 – L⊙ (1032 erg/ s): 1.86 4.67 4.64 6.94 4.71 2.49 1.81 0.82 0.17 – Fig. 2.3. The transmission characteristics of Johnson UBV and Kron Cousins RI filter systems. [Based on data published in Bessell (1990)] photometric systems. A well-known example is the standard UBV system originally introduced by Johnson. The filter functions for this system are shown in Fig. 2.3. In general, a filter function can be characterized by an effective wavelength, λeff , and a characteristic bandwidth, usually quoted as a full width at half maximum (FWHM). The FWHM is defined as |λ1 − λ2 |, with FX (λ1 ) = FX (λ2 ) = half the peak value of FX (λ ). Table 2.1 lists λeff and the FWHM for the fil- ters of the standard UBVRI photometric system. In this system, the FWHM are all of order 10% or larger of the corresponding λeff. Such ‘broad-band photometry’ can be used to characterize the overall shape of the spectral energy distribution of an object with high efficiency. Alternatively, one can use ‘narrow-band photometry’ with much narrower filters to image objects in a particular emission line or to study its detailed SED properties. (b) Magnitude and Color For historical reasons, the flux of an astronomical object in the optical band (and also in the near-infrared and near-ultraviolet bands) is usually quoted in terms of apparent magnitude: mX = −2.5 log( fX / fX,0 ), (2.4) where the flux zero-point fX,0 has traditionally been taken as the flux in the X band of the bright star Vega. In recent years it has become more common to use ‘AB-magnitudes’, for which ! fX,0 = 3.6308 × 10−20 erg s−1 cm−2 Hz−1 FX (c/ν)dν. (2.5) Here ν is the frequency and c is the speed of light. Similarly, the luminosities of objects (in waveband X) are often quoted as an absolute magnitude: MX = −2.5 log(LX ) + CX , where CX  2.1 Astronomical Observations 29 is a zero-point. It is usually convenient to write LX in units of the solar luminosity in the same band, L⊙X. The values of L⊙X in the standard UBVRI photometric system are listed in Table 2.1. It then follows that " # LX MX = −2.5 log + M⊙X , (2.6) L⊙X where M⊙X is the absolute magnitude of the Sun in the waveband in consideration. Using Eq. (2.1), we have mX − MX = 5 log(r/r0 ), (2.7) where r0 is a fiducial distance at which mX and MX are defined to have the same value. Conven- tionally, r0 is chosen to be 10 pc (1 pc = 1 parsec = 3.0856 × 1018 cm; see §2.1.3 for a definition). According to this convention, the Vega absolute magnitudes of the Sun in the UBVRI photometric system have the values listed in Table 2.1. The quantity (mX − MX ) for an astronomical object is called its distance modulus. If we know both mX and MX for an object, then Eq. (2.7) can be used to obtain its distance. Conversely, if we know the distance to an object, a measurement of its apparent magnitude (or flux) can be used to obtain its absolute magnitude (or luminosity). Optical astronomers usually express surface brightness in terms of magnitudes per square arcsecond. In such ‘units’, the surface brightness in a band X is denoted by µX , and is related to the surface brightness in physical units, IX , according to " # IX µX = −2.5 log + 21.572 + M⊙X. (2.8) L⊙ pc−2 Note that it is the flux, not the magnitude, that is additive. Thus in order to obtain the total (apparent) magnitude from an image, one must first convert magnitude per unit area into flux per unit area, integrate the flux over the entire image, and then convert the total flux back to a total magnitude. If observations are made for an object in more than one waveband, then the difference between the magnitudes in any two different bands defines a color index (which corresponds to the slope of the SED between the two wavebands). For example, (B −V ) ≡ mB − mV = MB − MV (2.9) is called the (B −V ) color of the object. 2.1.2 Spectroscopy From spectroscopic observations one obtains spectra for objects, i.e. their SEDs fλ or fν defined so that fλ dλ and fν dν are the fluxes received in the elemental wavelength and frequency ranges dλ at λ and dν at ν. From the relation between wavelength and frequency, λ = c/ν, we then have that fν = λ 2 fλ /c and fλ = ν 2 fν /c. (2.10) At optical wavelengths, spectroscopy is typically performed by guiding the light from an object to a spectrograph where it is dispersed according to wavelength. For example, in multi-object fiber spectroscopy, individual objects are imaged onto the ends of optical fibers which take the light to prism or optical grating where it is dispersed. The resulting spectra for each individual fiber are then imaged on a detector. Such spectroscopy loses all information about the distri- bution of each object’s light within the circular aperture represented by the end of the fiber. In long-slit spectroscopy, on the other hand, the object of interest is imaged directly onto the spectrograph slit, resulting in a separate spectrum from each point of the object falling on the 30 Observational Facts slit. Finally, in an integral field unit (or IFU) the light from each point within the image of an extended object is led to a different point on the slit (for example, by optical fibers) result- ing in a three-dimensional data cube with two spatial dimensions and one dimension for the wavelength. At other wavelengths quite different techniques can be used to obtain spectral information. For example, at infrared and radio wavelengths the incoming signal from a source may be Fourier analyzed in time in order to obtain the power at each frequency, while at X-ray wavelengths the energy of each incoming photon can be recorded and the energies of different photons can be binned to obtain the spectrum. Spectroscopic observations can give us a lot of information which photometric observations cannot. A galaxy spectrum usually contains a slowly varying component called the continuum, with localized features produced by emission and absorption lines (see Fig. 2.12 below for some examples). It is a superposition of the spectra of all the individual stars in the galaxy, modified by emission and absorption from the gas and dust lying between the stars. From the ultraviolet through the near-infrared the continuum is due primarily to bound–free transitions in the pho- tospheres of the stars, in the mid- and far-infrared it is dominated by thermal emission from dust grains, in the radio it is produced by diffuse relativistic and thermal electrons within the galaxy, and in the X-ray it comes mainly from accretion of gas onto compact stellar remnants or a central black hole. Emission and absorption lines are produced by bound–bound transitions within atoms, ions and molecules, both in the outer photospheres of stars and in the interstellar gas. By analyzing a spectrum, we may infer the relative importance of these various processes, thereby understanding the physical properties of the galaxy. For example, the strength of a par- ticular emission line depends on the abundance of the excited state that produces it, which in turn depends not only on the abundance of the corresponding element but also on the temperature and ionization state of the gas. Thus emission line strengths can be used to measure the temperature, density and chemical composition of interstellar gas. Absorption lines, on the other hand, mainly arise in the atmospheres of stars, and their relative strengths contain useful information regard- ing the age and metallicity of the galaxy’s stellar population. Finally, interstellar dust gives rise to continuum absorption with broad characteristic features. In addition, since dust extinction is typically more efficient at shorter wavelengths, it also causes reddening, a change of the overall slope of the continuum emission. Spectroscopic observations have another important application. The intrinsic frequency of photons produced by electron transitions between two energy levels E1 and E2 is ν12 = (E2 − E1 )/hP , where hP is Planck’s constant, and we have assumed E2 > E1. Now suppose that these photons are produced by atoms moving with velocity v relative to the observer. Because of the Doppler effect, the observed photon frequency will be (assuming v ≪ c), " # v · r̂ νobs = 1 − ν12 , (2.11) c where r̂ is the unit vector of the emitting source relative to the observer. Thus, if the source is receding from the observer, the observed frequency is redshifted, νobs < ν12 ; conversely, if the source is approaching the observer, the observed frequency is blueshifted, νobs > ν12. It is convenient to define a redshift parameter to characterize the change in frequency, ν12 z≡ − 1. (2.12) νobs For the Doppler effect considered here, we have z = v · r̂/c. Clearly, by studying the properties of spectral lines from an object, one may infer the kinematics of the emitting (or absorbing) material.  2.1 Astronomical Observations 31 Fig. 2.4. (a) An illustration of the broadening of a spectral line by the velocity dispersion of stars in a stellar system. A telescope collects light from all stars within a cylinder through the stellar system. Each star contributes a narrow spectral line with rest frequency ν12 , which is Doppler shifted to a different frequency ν = ν12 + ∆ν due to its motion along the line-of-sight. The superposition of many such line profiles produces a broadened line, with the profile given by the convolution of the original stellar spectral line and the velocity distribution of the stars in the cylinder. (b) An illustration of long-slit spectroscopy of a thin rotating disk along the major axis of the $ image. In the plot, the rotation speed is assumed to depend on the distance from the center as Vrot (x) ∝ x/(1 + x2 ). As an example, suppose that the emitting gas atoms in an object have random motions along the line-of-sight drawn from a velocity distribution f (v ) dv. The observed photons will then have the following frequency distribution: F(νobs ) dνobs = f (v )(c/ν12 ) dνobs , (2.13) where v is related to νobs by v = c(1 − νobs /ν12 ), and we have neglected the natural width of atomic spectral lines. Thus, by observing F(νobs ) (the emission line profile in frequency space), we can infer f (v ). If the random motion is caused by thermal effects, we can infer the temperature of the gas from the observed line profile. For a stellar system (e.g. an ellip- tical galaxy) the observed spectral line is the convolution of the original stellar line profile S(ν) (which is a luminosity weighted sum of the spectra of all different stellar types that con- tribute to the flux) with the line-of-sight velocity distribution of all the stars in the observational aperture, ! F(νobs ) = S [νobs (1 + v /c)] f (v )dv. (2.14) Thus, each narrow, stellar spectral line is broadened by the line-of-sight velocity dispersion of the stars that contribute to that line (see Fig. 2.4a). If we know the type of stars that domi- nate the spectral lines in consideration, we can estimate%S(ν) and use the above relation to infer the properties % of f (v ), such as the mean velocity, v = v f (v ) dv , and the velocity dispersion, σ = [ (v − v )2 f (v ) dv ]1/2. Similarly, long-slit and IFU spectroscopy of extended objects can be used not only to study random motions along each line-of-sight through the source, but also to study large-scale flows in the source. An important example here is the rotation of galaxy disks. Suppose that the rotation of a disk around its axis is specified by a rotation curve, Vrot (R), which gives the rotation velocity as a function of distance to the disk center. Suppose further that the inclination angle between 32 Observational Facts the rotation axis and the line-of-sight is i. If we put a long slit along the major axis of the image of the disk, it is easy to show that the frequency shift along the slit is Vrot (R) sin i νobs (R) − ν12 = ± ν12 , (2.15) c where the + and − signs correspond to points on opposite sides of the disk center (see Fig. 2.4b). Thus the rotation curve of the disk can be measured from its long-slit spectrum and from its apparent shape (which allows the inclination angle to be estimated under the assumption that the disk is intrinsically round). 2.1.3 Distance Measurements A fundamental task in astronomy is the determination of the distances to astronomical objects. As we have seen above, the direct observables from an astronomical object are its angular size on the sky and its energy flux at the position of the observer. Distance is therefore required in order to convert these observables into physical quantities. In this subsection we describe the principles behind some of the most important methods for estimating astronomical distances. (a) Trigonometric Parallax The principle on which this distance measure is based is very sim- ple. We are all familiar with the following: when walking along one direction, nearby and distant objects appear to change their orientation with respect to each other. If the walked distance b is much smaller than the distance to an object d (assumed to be perpendicular to the direction of motion), then the change of the orientation of the object relative to an object at infinity is θ = b/d. Thus, by measuring b and θ we can obtain the distance d. This is called the trigono- metric parallax method, and can be used to measure distances to some relatively nearby stars. In principle, this can be done by measuring the change of the position of a star relative to one or more background objects (assumed to be at infinity) at two different locations. Unfortunately, the baseline provided by the Earth’s diameter is so short that even the closest stars do not have a measurable trigonometric parallax. Therefore, real measurements of stellar trigonometric paral- lax have to make use of the baseline provided by the diameter of the Earth’s orbit around the Sun. By measuring the trigonometric parallax, πt , which is half of the angular change in the position of a star relative to the background as measured over a six month interval, we can obtain the distance to the star as A d= , (2.16) tan(πt ) where A = 1 AU = 1.49597870 × 1013 cm is the length of the semimajor axis of the Earth’s orbit around the Sun. The distance corresponding to a trigonometric parallax of 1 arcsec is defined as 1 parsec (or 1 pc). From the Earth the accuracy with which πt can be measured is restricted by atmospheric seeing, which causes a blurring of the images. This problem is circumvented when using satellites. With the Hipparcos satellite reliable distances have been measured for nearby stars with πt > −3 ∼ 10 arcsec, or with distances d < ∼ 1 kpc. The GAIA satellite, which is currently scheduled for launch in 2012, will be able to measure parallaxes for stars with an accuracy of ∼ 2 × 10−4 arcsec, which will allow distance measurements to 10% accuracy for ∼ 2 × 108 stars. (b) Motion-Based Methods The principle of this distance measurement is also very simple. We all know that the angle subtended by an object of diameter l at a distance d is θ = l/d (assuming l ≪ d). If we measure the angular diameters of the same object from two distances, d1 and d2 , then the difference between them is ∆θ = l∆d/d 2 = θ ∆d/d, where ∆d = |d1 − d2 | is assumed to be much smaller than both d1 and d2 , and d = (d1 d2 )1/2 can be considered the distance to the object. Thus, we can estimate d by measuring ∆θ and ∆d. For a star cluster  2.1 Astronomical Observations 33 consisting of many stars, the change of its distance over a time interval ∆t is given by ∆d = vr ∆t, where vr is the mean radial velocity of the cluster and can be measured from the shift of its spectrum. If we can measure the change of the angular size of the cluster during the same time interval, ∆θ , then the distance to the cluster can be estimated from d = θ vr ∆t/∆θ. This is called the moving-cluster method. Another distance measure is based on the angular motion of cluster stars caused by their veloc- ity with respect to the Sun. If all stars in a star cluster had the same velocity, the extensions of their proper motion vectors would converge to a single point on the celestial sphere (just like the two parallel rails of a railway track appear to converge to a point at large distance). By mea- suring the proper motions of the stars in a star cluster, this convergent point can be determined. Because of the geometry, the line-of-sight from the observer to the convergent point is parallel to the velocity vector of the star cluster. Hence, the angle, φ , between the star cluster and its con- vergent point, which can be measured, is the same as that between the proper motion vector and its component along the line-of-sight between the observer and the star cluster. By measuring the cluster’s radial velocity vr , one can thus obtain the transverse velocity vt = vr tan φ. Comparing vt to the proper motion of the star cluster then yields its distance. This is called the convergent- point method and can be used to estimate accurate distances of star clusters up to a few hundred parsec. (c) Standard Candles and Standard Rulers As shown by Eqs. (2.1) and (2.2), the luminos- ity and physical size of an object are related through the distance to its flux and angular size, respectively. Since the flux and angular size are directly observable, we can estimate the distance to an object if its luminosity or its physical size can be obtained in a distance-independent way. Objects whose luminosities and physical sizes can be obtained in such a way are called standard candles and standard rulers, respectively. These objects play an important role in astronomy, not only because their distances can be determined, but more importantly, because they can serve as distance indicators to calibrate the relation between distance and redshift, allowing the distances to other objects to be determined from their redshifts, as we will see below. One important class of objects in cosmic distance measurements is the Cepheid variable stars (or Cepheids for short). These objects are observed to change their apparent magnitudes regu- larly, with periods ranging from 2 to 150 days. The period is tightly correlated with the star’s luminosity, such that M = −a − b log P, (2.17) where P is the period of light variation in days, and a and b are two constants which can be deter- mined using nearby Cepheids whose distances have been measured using another method. For example, using the trigonometric parallaxes of Cepheids measured with the Hipparcos satellite, Feast & Catchpole (1997) obtained the following relation between P and the absolute magnitude in the V band: MV = −1.43 − 2.81 log P, with a standard error in the zero-point of about 0.10 magnitudes (see Madore & Freedman, 1991, for more examples of such calibrations). Once the luminosity–period relation is calibrated, and if it is universally valid, it can be applied to distant Cepheids (whose distances cannot be obtained from trigonometric parallax or proper motion) to obtain their distances from measurements of their variation periods. Since Cepheids are relatively bright, with absolute magnitudes MV ∼ −3, telescopes with sufficiently high spatial resolution, such as the Hubble Space Telescope (HST), allow Cepheid distances to be determined for objects out to ∼ 10 Mpc. Another important class of objects for distance measurements are Type Ia supernovae (SNIa), which are exploding stars with well-calibrated light profiles. Since these objects can reach peak luminosities up to ∼ 1010 L⊙ (so that they can outshine an entire galaxy), they can be observed out to cosmological distances of several thousand megaparsecs. Empirically it has been found 34 Observational Facts that the peak luminosities of SNIa are remarkably similar (e.g. Branch & Tammann, 1992). In fact, there is a small dispersion in peak luminosities, but this has been found to be correlated with the rate at which the luminosity decays and so can be corrected (e.g. Phillips et al., 1999). Thus, one can obtain the relative distances to Type Ia supernovae by measuring their light curves. The absolute distances can then be obtained once the absolute values of the light curves of some nearby Type Ia supernovae are calibrated using other (e.g. Cepheid) distances. As we will see in §2.10.1, SNIa play an important role in constraining the large scale geometry of the Universe. (d) Redshifts as Distances One of the most important discoveries in modern science was Hubble’s (1929) observation that almost all galaxies appear to move away from us, and that their recession velocities increase in direct proportion to their distances from us, vr ∝ r. This relation, called the Hubble law, is explained most naturally if the Universe as a whole is assumed to be expanding. If the expansion is homogeneous and isotropic, then the distance between any two objects comoving with the expanding background can be written as r(t) = a(t)r(t ′ )/a(t ′ ), where a(t) is a time-dependent scale factor of the Universe, describing the expansion. It then follows that the relative separation velocity of the objects is vr = ṙ = H(t)r, where H(t) ≡ ȧ(t)/a(t). (2.18) This relation applied at the present time gives vr = H0 r, as observed by Hubble. Since the reces- sion velocity of an object can be measured from its redshift z, the distance to the object simply follows from r = c z/H0 (assuming vr ≪ c). In practice, the object under consideration may move relative to the background with some (gravitationally induced) peculiar velocity, vpec , so that its observed velocity is the sum of this peculiar velocity along the line-of-sight, vpec,r , and the velocity due to the Hubble expansion: vr = H0 r + vpec,r. (2.19) In this case, the redshift is no longer a precise measurement of the distance, unless vpec,r ≪ H0 r. Since for galaxies the typical value for vpec is a few hundred kilometers per second, redshifts can be used to approximate distances for cz ≫ 1000 km s−1. In order to convert redshifts into distances, we need a value for the Hubble constant, H0. This can be obtained if the distances to some sufficiently distant objects can be measured indepen- dently of their redshifts. As mentioned above, such objects are called distance indicators. For many years, the value of the Hubble constant was very uncertain, with estimates ranging from ∼ 50 km s−1 Mpc−1 to ∼ 100 km s−1 Mpc−1 (current constraints on H0 are discussed in §2.10.1). To parameterize this uncertainty in H0 it has become customary to write H0 = 100h km s−1 Mpc−1 , (2.20) and to express all quantities that depend on redshift-based distances in terms of the reduced Hubble constant h. For example, distance determinations based on redshifts often contain a factor of h−1 , while luminosities based on these distances contain a factor h−2 , etc. If these factors are not present, it means that a specific value for the Hubble constant has been assumed, or that the distances were not based on measured redshifts. 2.2 Stars As we will see in §2.3, the primary visible constituent of most galaxies is the combined light from their stellar population. Clearly, in order to understand galaxy formation and evolution it is important to know the main properties of stars. In Table 2.1 we list some of the photomet- ric properties of the Sun. These, as well as the Sun’s mass and radius, M⊙ = 2 × 1033 g and  2.2 Stars 35 Table 2.2. Solar abundances in number relative to hydrogen. Element: H He C N O Ne Mg Si Fe (N/NH ) × 105 : 105 9800 36.3 11.2 85.1 12.3 3.80 3.55 4.68 Fig. 2.5. Spectra for stars of different spectral types. fλ is the flux per angstrom, and an arbitrary constant is added to each spectrum to avoid confusion. [Based on data kindly provided by S. Charlot] R⊙ = 7 × 1010 cm, are usually used as fiducial values when describing other stars. The abun- dance by number of some of the chemical elements in the solar system is given in Table 2.2. The fraction in mass of elements heavier than helium is referred to as the metallicity and is denoted by Z, and our Sun has Z⊙ ≈ 0.02. The relative abundances in a star are usually specified relative to those in the Sun: & ' (nA /nB )⋆ [A/B] ≡ log , (2.21) (nA /nB )⊙ where (nA /nB )⋆ is the number density ratio between element A and element B in the star, and (nA /nB )⊙ is the corresponding ratio for the Sun. Since all stars, except a few nearby ones, are unresolved (i.e. they appear as point sources), the only intrinsic properties that are directly observable are their luminosities, colors and spec- tra. These vary widely (some examples of stellar spectra are shown in Fig. 2.5) and form the basis for their classification. The most often used classification scheme is the Morgan–Keenan (MK) system, summarized in Tables 2.3 and 2.4. These spectral classes are further divided into decimal subclasses [e.g. from B0 (early) to B9 (late)], while luminosity classes are divided into subclasses such as Ia, Ib, etc. The importance of this classification is that, although entirely based 36 Observational Facts Table 2.3. MK spectral classes. Class Temperature Spectral characteristics O 28,000–50,000 K Hot stars with He II absorption; strong UV continuum B 10,000–28,000 K He I absorption; H developing in later classes A 7,500–10,000 K Strong H lines for A0, decreasing thereafter; Ca II increasing F 6,000–7,500 K Ca II stronger; H lines weaker; metal lines developing G 5,000–6,000 K Ca II strong; metal lines strong; H lines weaker K 3,500–5,000 K Strong metal lines; CH and CN developing; weak blue continuum M 2,500–3,500 K Very red; TiO bands developing strongly Table 2.4. MK luminosity classes. I Supergiants II Bright giants III Normal giants IV Subgiants V Dwarfs (main-sequence stars) on observable properties, it is closely related to the basic physical properties of stars. For exam- ple, the luminosity classes are related to surface gravities, while the spectral classes are related to surface temperatures (see e.g. Cox, 2000). Fig. 2.6 shows the color–magnitude relation of a large number of stars for which accurate distances are available (so that their absolute magnitudes can be determined). Such a diagram is called a Hertzsprung–Russell diagram (abbreviated as H-R diagram), and features predominantly in studies of stellar astrophysics. The MK spectral and luminosity classes are also indicated. Clearly, stars are not uniformly distributed in the color–magnitude space, but lie in several well- defined sequences. Most of the stars lie in the ‘main sequence’ (MS) which runs from the lower- right to the upper-left. Such stars are called main-sequence stars and have MK luminosity class V. The position of a star in this sequence is mainly determined by its mass. Above the main sequence one finds the much rarer but brighter giants, making up the MK luminosity classes I to IV, while the lower-left part of the H-R diagram is occupied by white dwarfs. The Sun, whose MK type is G2V, lies in the main sequence with V -band absolute magnitude 4.8 and (atmospheric) temperature 5780K. As a star ages it moves off the MS and starts to traverse the H-R diagram. The location of a star in the H-R diagram as function of time is called its evolutionary track which, again, is determined mainly by its mass. An important property of a stellar population is therefore its initial mass function (IMF), which specifies the abundance of stars as function of their initial mass (i.e. the mass they have at the time when reach the MS shortly after their formation). For a given IMF, and a given star-formation history, one can use the evolutionary tracks to predict the abundance of stars in the H-R diagram. Since the spectrum of a star is directly related to its position in the H-R diagram, this can be used to predict the spectrum of an entire galaxy, a procedure which is called spectral synthesis modeling. Detailed calculations of stellar evolution models (see Chapter 10) show that a star like our Sun has a MS lifetime of about 10 Gyr, and that the MS lifetime scales with mass roughly as M −3 , i.e. more massive (brighter) stars spend less time on the MS. This strong dependence of MS lifetime on mass has important observational consequences, because it implies that the spectrum of a stellar system (a galaxy) depends on its star-formation history. For a system where the current star-formation rate is high, so that many young massive stars are still on the main sequence, the stellar spectrum is expected to have a strong blue continuum produced by O and B stars. On the other hand, for a system where star  2.3 Galaxies 37 Fig. 2.6. The color–magnitude diagram (i.e. the H-R diagram) of 22,000 stars from the Hipparcos Cata- logue together with 1,000 low-luminosity stars (red and white dwarfs) from the Gliese Catalogue of Nearby Stars. The MK spectral and luminosity classes are also indicated, as are the luminosities in solar units. [Diagram from R. Powell, taken from Wikipedia] formation has been terminated a long time ago, so that all massive stars have already evolved off the MS, the spectrum (now dominated by red giants and the low-mass MS stars) is expected to be red. 2.3 Galaxies Galaxies, whose formation and evolution is the main topic of this book, are the building blocks of the Universe. They not only are the cradles for the formation of stars and metals, but also serve as beacons that allow us to probe the geometry of space-time. Yet it is easy to forget that it was not until the 1920s, with Hubble’s identification of Cepheid variable stars in the Andromeda Nebula, that most astronomers became convinced that the many ‘nebulous’ objects cataloged by John Dreyer in his 1888 New General Catalogue of Nebulae and Clusters of Stars and the two supplementary Index Catalogues are indeed galaxies. Hence, extragalactic astronomy is a relatively new science. Nevertheless, as we will see, we have made tremendous progress: we 38 Observational Facts Fig. 2.7. Examples of different types of galaxies. From left to right and top to bottom, NGC 4278 (E1), NGC 3377 (E6), NGC 5866 (SO), NGC 175 (SBa), NGC 6814 (Sb), NGC 4565 (Sb, edge on), NGC 5364 (Sc), Ho II (Irr I), NGC 520 (Irr II). [All images are obtained from the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration] have surveyed the local population of galaxies in exquisite detail covering the entire range of wavelengths, we have constructed redshift surveys with hundreds of thousands of galaxies to probe the large scale structure of the Universe, and we have started to unveil the population of galaxies at high redshifts, when the Universe was only a small fraction of its current age. 2.3.1 The Classification of Galaxies Fig. 2.7 shows a collage of images of different kinds of galaxies. Upon inspection, one finds that some galaxies have smooth light profiles with elliptical isophotes, others have spiral arms together with an elliptical-like central bulge, and still others have irregular or peculiar morpholo- gies. Based on such features, Hubble ordered galaxies in a morphological sequence, which is  2.3 Galaxies 39 (Normal spirals) (Ellipticals) Sa Sb Sc Im E0 E3 E6 S0 SBa SBb SBc IBm (Barred spirals) Fig. 2.8. A schematic representation of the Hubble sequence of galaxy morphologies. [Courtesy of R. Abraham; see Abraham (1998)] now referred to as the Hubble sequence or Hubble tuning-fork diagram (see Fig. 2.8). Hubble’s scheme classifies galaxies into four broad classes: (i) Elliptical galaxies: These have smooth, almost elliptical isophotes and are divided into subtypes E0, E1,... , E7, where the integer is the one closest to 10 (1 − b/a), with a and b the lengths of the semimajor and semiminor axes. (ii) Spiral galaxies: These have thin disks with spiral arm structures. They are divided into two branches, barred spirals and normal spirals, according to whether or not a recogniz- able bar-like structure is present in the central part of the galaxy. On each branch, galaxies are further divided into three classes, a, b and c, according to the following three criteria: the fraction of the light in the central bulge; the tightness with which the spiral arms are wound; the degree to which the spiral arms are resolved into stars, HII regions and ordered dust lanes. These three criteria are correlated: spirals with a pronounced bulge component usually also have tightly wound spiral arms with relatively faint HII regions, and are classified as Sa. On the other hand, spirals with weak or absent bulges usually have open arms and bright HII regions and are classified as Sc. When the three criteria give conflicting indications, Hubble put most emphasis on the openness of the spiral arms. (iii) Lenticular or S0 galaxies: This class is intermediate between ellipticals and spirals. Like ellipticals, lenticulars have a smooth light distribution with no spiral arms or HII regions. Like spirals they have a thin disk and a bulge, but the bulge is more dominant than that in a spiral galaxy. They may also have a central bar, in which case they are classified as SB0. (iv) Irregular galaxies: These objects have neither a dominating bulge nor a rotationally sym- metric disk and lack any obvious symmetry. Rather, their appearance is generally patchy, dominated by a few HII regions. Hubble did not include this class in his original sequence because he was uncertain whether it should be considered an extension of any of the other classes. Nowadays irregulars are usually included as an extension to the spiral galaxies. Ellipticals and lenticulars together are often referred to as early-type galaxies, while the spirals and irregulars make up the class of late-type galaxies. Indeed, traversing the Hubble sequence from the left to the right the morphologies are said to change from early- to late-type. Although somewhat confusing, one often uses the terms ‘early-type spirals’ and ‘late-type spirals’ to refer to galaxies at the left or right of the spiral sequence. We caution, though, that this historical nomenclature has no direct physical basis: the reference to ‘early’ or ‘late’ should not be interpreted as reflecting a property of the galaxy’s evolutionary state. Another largely historical 40 Observational Facts Table 2.5. Galaxy morphological types. Hubble E E-SO SO SO-Sa Sa Sa-b Sb Sb-c Sc Sc-Irr Irr deV E SO− SO0 SO+ Sa Sab Sb Sbc Scd Sdm Im T −5 −3 −2 0 1 2 3 4 6 8 10 Fig. 2.9. Fractional luminosity of the spheroidal bulge component in a galaxy as a function of morphologi- cal type (based on the classification of de Vaucouleurs). Data points correspond to individual galaxies, and the curve is a fit to the mean. Elliptical galaxies (Type = −5) are considered to be pure bulges. [Based on data presented in Simien & de Vaucouleurs (1986)] nomenclature, which can be confusing at times, is to refer to faint galaxies with MB > ∼ −18 as ‘dwarf galaxies’. In particular, early-type dwarfs are often split into dwarf ellipticals (dE) and dwarf spheroidals (dSph), although there is no clear distinction between these types – often the term dwarf spheroidals is simply used to refer to early-type galaxies with MB > ∼ −14. Since Hubble, a variety of other classification schemes have been introduced. A commonly used one is due to de Vaucouleurs (1974). He put spirals in the Hubble sequence into a finer gradation by adding new types such as SOa, Sab, Sbc (and the corresponding barred types). After finding that many of Hubble’s irregular galaxies in fact had weak spiral arms, de Vaucouleurs also extended the spiral sequence to irregulars, adding types Scd, Sd, Sdm, Sm, Im and I0, in order of decreasing regularity. (The m stands for ‘Magellanic’ since the Magellanic Clouds are the prototypes of this kind of irregulars.) Furthermore, de Vaucouleurs used numbers between −6 and 10 to represent morphological types (the de Vaucouleurs’ T types). Table 2.5 shows the correspondence between de Vaucouleurs’ notations and Hubble’s notations – note that the numerical T types do not distinguish between barred and unbarred galaxies. As shown in Fig. 2.9, the morphology sequence according to de Vaucouleurs’ classification is primarily a sequence in the importance of the bulge. The Hubble classification and its revisions encompass the morphologies of the majority of the observed galaxies in the local Universe. However, there are also galaxies with strange appearances which defy Hubble’s classification. From their morphologies, these ‘peculiar’  2.3 Galaxies 41 Fig. 2.10. The peculiar galaxy known as the Antennae, a system exhibiting prominent tidal tails (the left inlet), a signature of a recent merger of two spiral galaxies. The close-up of the center reveals the presence of large amounts of dust and many clusters of newly formed stars. [Courtesy of B. Whitmore, NASA, and Space Telescope Science Institute] galaxies all appear to have been strongly perturbed in the recent past and to be far from dynam- ical equilibrium, indicating that they are undergoing a transformation. A good example is the Antennae (Fig. 2.10) where the tails are produced by the interaction of the two spiral galaxies, NGC 4038 and NGC 4039, in the process of merging. The classifications discussed so far are based only on morphology. Galaxies can also be clas- sified according to other properties. For instance, they can be classified into bright and faint according to luminosity, into high and low surface brightness according to surface brightness, into red and blue according to color, into gas-rich and gas-poor according to gas content, into quiescent and starburst according to their current level of star formation, and into normal and active according to the presence of an active nucleus. All these properties can be measured obser- vationally, although often with some difficulty. An important aspect of the Hubble sequence (and its modifications) is that many of these properties change systematically along the sequence (see Figs. 2.11 and 2.12), indicating that it reflects a sequence in the basic physical properties of galax- ies. However, we stress that the classification of galaxies is far less clear cut than that of stars, whose classification has a sound basis in terms of the H-R diagram and the evolutionary tracks. 2.3.2 Elliptical Galaxies Elliptical galaxies are characterized by smooth, elliptical surface brightness distributions, contain little cold gas or dust, and have red photometric colors, characteristic of an old stellar pop- ulation. In this section we briefly discuss some of the main, salient observational properties. A more in-depth discussion, including an interpretation within the physical framework of galaxy formation, is presented in Chapter 13. (a) Surface Brightness Profiles The one-dimensional surface brightness profile, I(R), of an elliptical galaxy is usually defined as the surface brightness as a function of the isophotal 42 Observational Facts 11 1.5 log 〈LB〉 log 〈R25〉 10 1 9 0.5 12 11 log 〈MHI〉 log 〈MT〉 11 10 10 9 2.6 0 2.4 log 〈MHI/LB〉 log 〈ΣT〉 2.2 –1 2 –2 1.8 1 1.5 0.8 1 〈(B–V)〉 log 〈ΣHI〉 0.5 0.6 0 0.4 –0.5 0.2 S0 Sa Sb Sc Sd Im S0 Sa Sb Sc Sd Im E S0a Sab Sbc Scd Sm E S0a Sab Sbc Scd Sm Fig. 2.11. Galaxy properties along the Hubble morphological sequence based on the RC3-UGC sample. Filled circles are medians, open ones are mean values. The bars bracket the 25 and 75 percentiles. Properties plotted are LB (blue luminosity in L⊙ ), R25 (the radius in kpc of the 25mag arcsec−2 isophote in the B-band), MT (total mass in solar units within a radius R25 /2), MHI (HI mass in solar units), MHI /LB , ΣT (total mass surface density), ΣHI (HI mass surface density), and the B − V color. [Based on data presented in Roberts & Haynes (1994)] semimajor axis length R. If the position angle of the semimajor axis changes with radius, a phenomenon called isophote twisting, then I(R) traces the surface brightness along a curve that connects the intersections of each isophote with its own major axis. The surface brightness profile of spheroidal galaxies is generally well fit by the Sérsic profile (Sérsic, 1968), or R1/n profile,1 ( " #1/n ) ( *" # +) R R 1/n I(R) = I0 exp −βn = Ie exp −βn −1 , (2.22) Re Re where I0 is the central surface brightness, n is the so-called Sérsic index which sets the concen- tration of the profile, Re is the effective radius that encloses half of the total light, and Ie = I(Re ). Surface brightness profiles are often expressed in terms of µ ∝ −2.5 log(I) (which has the units of mag arcsec−2 ), for which the Sérsic profile takes the form (" # ) R 1/n µ (R) = µe + 1.086 βn −1. (2.23) Re 1 A similar formula, but with R denoting 3-D rather than projected radius, was used by Einasto (1965) to describe the stellar halo of the Milky Way.  2.3 Galaxies 43 Fig. 2.12. Spectra of different types of galaxies from the ultraviolet to the near-infrared. From ellipticals to late-type spirals, the blue continuum and emission lines become systematically stronger. For early-type galaxies, which lack hot, young stars, most of the light emerges at the longest wavelengths, where one sees absorption lines characteristic of cool K stars. In the blue, the spectrum of early-type galaxies show strong H and K absorption lines of calcium and the G band, characteristic of solar type stars. Such galaxies emit little light at wavelengths shorter than 4000 Å and have no emission lines. In contrast, late-type galaxies and starbursts emit most of their light in the blue and near-ultraviolet. This light is produced by hot young stars, which also heat and ionize the interstellar medium giving rise to strong emission lines. [Based on data kindly provided by S. Charlot] The value for βn follows from the definition of Re and is well approximated by βn = 2n − 0.324 ∼ 1). Note that Eq. (2.22) reduces to a simple exponential profile for n = 1. The (but only for n > total luminosity of a spherical system with a Sérsic profile is ! ∞ 2π n Γ(2n) L = 2π I(R) R dR = I0 R2e , (2.24) 0 (βn )2n with Γ(x) the gamma function. Early photometry of the surface brightness profiles of normal giant elliptical galaxies was well fit by a de Vaucouleurs profile, which is a Sérsic profile with n = 4 (and βn = 7.67) and is therefore also called a R1/4 -profile. With higher accuracy photometry and with measurements of higher and lower luminosity galaxies, it became clear that ellipticals as a class are better fit by the more general Sérsic profile. In fact, the best-fit values for n have 44 Observational Facts Fig. 2.13. Correlation between the Sérsic index, n, and the absolute magnitude in the B-band for a sample of elliptical galaxies. The vertical dotted lines correspond to MB = −18 and MB = −20.5 and are shown to facilitate a comparison with Fig. 2.14. [Data compiled and kindly made available by A. Graham (see Graham & Guzmán, 2003)] Fig. 2.14. The effective radius (left panel) and the average surface brightness within the effective radius (right panel) of elliptical galaxies plotted against their absolute magnitude in the B-band. The vertical dotted lines correspond to MB = −18 and MB = −20.5. [Data compiled and kindly made available by A. Graham (see Graham & Guzmán, 2003), combined with data taken from Bender et al. (1992)] been found to be correlated with the luminosity and size of the galaxy: while at the faint end dwarf ellipticals have best-fit values as low as n ∼ 0.5, the brightest ellipticals can have Sérsic indices n > ∼ 10 (see Fig. 2.13). Instead of I0 or Ie , one often characterizes the surface brightness of an elliptical galaxy via the average surface brightness within the effective radius, ⟨I⟩e = L/(2π R2e ), or, in magnitudes, ⟨µ ⟩e. Fig. 2.14 shows how Re and ⟨µ ⟩e are correlated with luminosity. At the bright end (MB < ∼ −18), the sizes of elliptical galaxies increase strongly with luminosity. Consequently, the average sur- face brightness actually decreases with increasing luminosity. At the faint end (MB > ∼ −18),  2.3 Galaxies 45 however, all ellipticals have roughly the same effective radius (Re ∼ 1 kpc), so that the average surface brightness increases with increasing luminosity. Because of this apparent change-over in properties, ellipticals with MB > ∼ −18 are typically called ‘dwarf’ ellipticals, in order to distin- guish them from the ‘normal’ ellipticals (see §2.3.5). However, this alleged ‘dichotomy’ between dwarf and normal ellipticals has recently been challenged. A number of studies have argued that there is actually a smooth and continuous sequence of increasing surface brightness with increas- ing luminosity, except for the very bright end (MB < ∼ −20.5) where this trend is reversed (e.g. Jerjen & Binggeli, 1997; Graham & Guzmán, 2003). The fact that the photometric properties of elliptical galaxies undergo a transition around MB ∼ −20.5 is also evident from their central properties (in the inner few hundred parsec). High spatial resolution imaging with the HST has revealed that the central surface brightness profiles of elliptical galaxies are typically not well described by an inward extrapolation of the Sérsic pro- ∼ −20.5 typically have a deficit in I(R) files fit to their outer regions. Bright ellipticals with MB < with respect to the best-fit Sérsic profile, while fainter ellipticals reveal excess surface bright- ness. Based on the value of the central cusp slope γ ≡ d log I/d log r the population of ellipticals has been split into ‘core’ (γ < 0.3) and ‘power-law’ (γ ≥ 0.3) systems. The majority of bright galaxies with MB < ∼ −20.5 have cores, while power-law galaxies typically have MB > −20.5 (Ferrarese et al., 1994; Lauer et al., 1995). Early results, based on relatively small samples, sug- gested a bimodal distribution in γ , with virtually no galaxies in the range 0.3 < γ < 0.5. However, subsequent studies have significantly weakened the evidence for a clear dichotomy, finding a population of galaxies with intermediate properties (Rest et al., 2001; Ravindranath et al., 2001). In fact, recent studies, using significantly larger samples, have argued for a smooth transition in nuclear properties, with no evidence for any dichotomy (Ferrarese et al., 2006b; Côté et al., 2007; see also §13.1.2). (b) Isophotal Shapes The isophotes of elliptical galaxies are commonly fitted by ellipses and characterized by their minor-to-major axis ratios b/a (or, equivalently, by their ellipticities ε = 1 − b/a) and by their position angles. In general, the ellipticity may change across the system, in which case the overall shape of an elliptical is usually defined by some characteristic ellipticity (e.g. that of the isophote which encloses half the total light). In most cases, however, the variation of ε with radius is not large, so that the exact definition is of little consequence. For normal elliptical galaxies the axis ratio lies in the range 0.3 < ∼ b/a ≤ 1, corresponding to types E0 to E7. In addition to the ellipticity, the position angle of the isophotes may also change with radius, a phenomenon called isophote twisting. Detailed modeling of the surface brightness of elliptical galaxies shows that their isophotes are generally not exactly elliptical. The deviations from perfect ellipses are conveniently quantified by the Fourier coefficients of the function ∞ ∆(φ ) ≡ Riso (φ ) − Rell (φ ) = a0 + ∑ (an cos nφ + bn sin nφ ) , (2.25) n=1 where Riso (φ ) is the radius of the isophote at angle φ and Rell (φ ) is the radius of an ellipse at the same angle (see Fig. 2.15). Typically one considers the ellipse that best fits the isophote in question, so that a0 , a1 , a2 , b1 and b2 are all consistent with zero within the errors. The deviations from this best-fit isophote are then expressed by the higher-order Fourier coefficients an and bn with n ≥ 3. Of particular importance are the values of the a4 coefficients, which indicate whether the isophotes are ‘disky’ (a4 > 0) or ‘boxy’ (a4 < 0), as illustrated in Fig. 2.15. The diskiness of an isophote is defined as the dimensionless quantity, a4 /a, where a is the length of the semimajor axis of the isophote’s best-fit ellipse. We caution that some authors use an alternative method to specify the deviations of isophotes from pure ellipses. Instead of using 46 Observational Facts Fig. 2.15. An illustration of boxy and disky isophotes (solid curves). The dashed curves are the corresponding best-fit ellipses. isophote deviation from an ellipse, they quantify how the intensity fluctuates along the best-fit ellipse: ∞ I(φ ) = I0 + ∑ (An cos nφ + Bn sin nφ ) , (2.26) n=1 with I0 the intensity of the best-fit ellipse. The coefficients An and Bn are (approximately) related to an and bn according to , , , , , dI , , dI , , An = an , , , , Bn = bn ,, ,, , (2.27) dR dR √ where R = a 1 − ε , with ε the ellipticity of the best-fit ellipse. The importance of the disky/boxy classification is that boxy and disky ellipticals turn out to have systematically different properties. Boxy ellipticals are usually bright, rotate slowly, and show stronger than average radio and X-ray emission, while disky ellipticals are fainter, have significant rotation and show little or no radio and X-ray emission (e.g. Bender et al., 1989; Pasquali et al., 2007). In addition, the diskiness is correlated with the nuclear properties as well; disky ellipticals typically have steep cusps, while boxy ellipticals mainly harbor central cores (e.g. Jaffe et al., 1994; Faber et al., 1997). (c) Colors Elliptical galaxies in general have red colors, indicating that their stellar contents are dominated by old, metal-rich stars (see §10.3). In addition, the colors are tightly correlated with the luminosity such that brighter ellipticals are redder (Sandage & Visvanathan, 1978). As we will see in §13.5, the slope and (small) scatter of this color–magnitude relation puts tight constraints on the star-formation histories of elliptical galaxies. Ellipticals also display color gradient. In general, the outskirt has a bluer color than the central region. Peletier et al. (1990) obtained a mean logarithmic gradient of ∆(U − R)/∆ log r = −0.20 ± 0.02 mag in U − R, and of ∆(B − R)/∆ log r = −0.09 ± 0.02 mag in B − R, in good agreement with the results obtained by Franx et al. (1989b). (d) Kinematic Properties Giant ellipticals generally have low rotation velocities. Observa- tionally, this may be characterized by the ratio of maximum line-of-sight streaming motion vm (relative to the mean velocity of the galaxy) to σ , the average value of the line-of-sight velocity dispersion interior to ∼ Re /2. This ratio provides a measure of the relative importance of ordered and random motions within the galaxy. For $ isotropic, oblate galaxies flattened by the centrifu- gal force generated by rotation, vm /σ ≈ ε /(1 − ε ), with ε the ellipticity of the spheroid (see §13.1.7). As shown in Fig. 2.16a, for bright ellipticals, vm /σ lies well below this prediction,  2.3 Galaxies 47 Fig. 2.16. (a) The ratio vm /σ for ellipticals and bulges (with bulges marked by horizontal bars) versus ellip- ticity. Open circles are for bright galaxies with MB ≤ 20.5, with upper limits marked by downward arrows; solid circles are for early types with −20.5 < MB < −18. The solid curve is the relation expected for an oblate galaxy flattened by rotation. [Based on data published in Davies et al. (1983)] (b) The rotation param- eter (v /σ )∗ (defined as the ratio of vm /σ to the value expected for an isotropic oblate spheroid flattened purely by rotation) versus the average diskiness of the galaxy. [Based on data published in Kormendy & Bender (1996)] indicating that their flattening must be due to velocity anisotropy, rather than rotation. In con- trast, ellipticals of intermediate luminosities (with absolute magnitude −20.5 < ∼ MB < ∼ −18.0) and spiral bulges have vm /σ values consistent with rotational flattening. Fig. 2.16b shows, as noted above, that disky and boxy ellipticals have systematically different kinematics: while disky ellipticals are consistent with rotational flattening, rotation in boxy ellipticals is dynamically unimportant. When the kinematic structure of elliptical galaxies is examined in more detail a wide range of behavior is found. In most galaxies the line-of-sight velocity dispersion depends only weakly on position and is constant or falls at large radii. Towards the center the dispersion may drop weakly, remain flat, or rise quite sharply. The behavior of the mean line-of-sight streaming velocity is even more diverse. While most galaxies show maximal streaming along the major axis, a sub- stantial minority show more complex behavior. Some have non-zero streaming velocities along the minor axis, and so it is impossible for them to be an oblate body rotating about its sym- metry axis. Others have mean motions which change suddenly in size, in axis, or in sign in the inner regions, the so-called kinematically decoupled cores. Such variations point to a variety of formation histories for apparently similar galaxies. At the very center of most nearby ellipticals (and also spiral and S0 bulges) the velocity dis- persion is observed to rise more strongly than can be understood as a result of the gravitational effects of the observed stellar populations alone. It is now generally accepted that this rise sig- nals the presence of a central supermassive black hole. Such a black hole appears to be present in virtually every galaxy with a significant spheroidal component, and to have a mass which is roughly 0.1% of the total stellar mass of the spheroid (Fig. 2.17). A more detailed discussion of supermassive black holes is presented in §13.1.4. (e) Scaling Relations The kinematic and photometric properties of elliptical galaxies are cor- related. In particular, ellipticals with a larger (central) velocity dispersion are both brighter, known as the Faber–Jackson relation, and larger, known as the Dn -σ relation (Dn is the isophotal 48 Observational Facts Fig. 2.17. The masses of central black holes in ellipticals and spiral bulges plotted against the absolute magnitude (left) and velocity dispersion (right) of their host spheroids. [Adapted from Kormendy (2001)] log R e (kpc) log σ0 (km/s) Fig. 2.18. The fundamental plane of elliptical galaxies in the log Re -log σ0 -⟨µ ⟩e space (σ0 is the central velocity dispersion, and ⟨µ ⟩e is the mean surface brightness within Re expressed in magnitudes per square arcsecond). [Plot kindly provided by R. Saglia, based on data published in Saglia et al. (1997) and Wegner et al. (1999)] diameter within which the average, enclosed surface brightness is equal to a fixed value). Fur- thermore, when plotted in the three-dimensional space spanned by log σ0 , log Re and log⟨I⟩e , elliptical galaxies are concentrated in a plane (see Fig. 2.18) known as the fundamental plane. In mathematical form, this plane can be written as log Re = a log σ0 + b log⟨I⟩e + constant, (2.28)  2.3 Galaxies 49 where ⟨I⟩e is the mean surface brightness within Re (not to be confused with Ie , which is the surface brightness at Re ). The values of a and b have been estimated in various photometric bands. For example, Jørgensen et al. (1996) obtained a = 1.24 ± 0.07, b = −0.82 ± 0.02 in the optical, while Pahre et al. (1998) obtained a = 1.53 ± 0.08, b = −0.79 ± 0.03 in the near-infrared. More recently, using 9,000 galaxies from the Sloan Digital Sky Survey (SDSS), Bernardi et al. (2003b) found the best fitting plane to have a = 1.49±0.05 and b = −0.75±0.01 in the SDSS r-band with a rms of only 0.05. The Faber–Jackson and Dn -σ relations are both two-dimensional projections of this fundamental plane. While the Dn -σ projection is close to edge-on and so has relatively little scatter, the Faber–Jackson projection is significantly tilted resulting in somewhat larger scatter. These relations can not only be used to determine the distances to elliptical galaxies, but are also important for constraining theories for their formation (see §13.4). (f) Gas Content Although it was once believed that elliptical galaxies contain neither gas nor dust, it has become clear over the years that they actually contain a significant amount of inter- stellar medium which is quite different in character from that in spiral galaxies (e.g. Roberts et al., 1991; Buson et al., 1993). Hot (∼ 107 K) X-ray emitting gas usually dominates the interstellar medium (ISM) in luminous ellipticals, where it can contribute up to ∼ 1010 M⊙ to the total mass of the system. This hot gas is distributed in extended X-ray emitting atmospheres (Fabbiano, 1989; Mathews & Brighenti, 2003), and serves as an ideal tracer of the gravitational potential in which the galaxy resides (see §8.2). In addition, many ellipticals also contain small amounts of warm ionized (104 K) gas as well as cold (< 100 K) gas and dust. Typical masses are 102 –104 M⊙ in ionized gas and 106 –108 M⊙ in the cold component. Contrary to the case for spirals, the amounts of dust and of atomic and molecular gas are not correlated with the luminosity of the elliptical. In many cases, the dust and/or ionized gas is located in the center of the galaxy in a small disk component, while other ellipticals reveal more complex, filamentary or patchy dust morphologies (e.g. van Dokkum & Franx, 1995; Tran et al., 2001). This gas and dust either results from accumulated mass loss from stars within the galaxy or has been accreted from external systems. The latter is supported by the fact that the dust and gas disks are often found to have kinematics decoupled from that of the stellar body (e.g. Bertola et al., 1992). 2.3.3 Disk Galaxies Disk galaxies have a far more complex morphology than ellipticals. They typically consist of a thin, rotationally supported disk with spiral arms and often a bar, plus a central bulge component. The latter can dominate the light of the galaxy in the earliest types and may be completely absent in the latest types. The spiral structure is best seen in face-on systems and is defined primarily by young stars, HII regions, molecular gas and dust absorption. Edge-on systems, on the other hand, give a better handle on the vertical structure of the disk, which often reveals two separate components: a thin disk and a thick disk. In addition, there are indications that disk galaxies also contain a spheroidal, stellar halo, extending out to large radii. In this subsection we briefly summarize the most important observational characteristics of disk galaxies. A more in-depth discussion, including models for their formation, is presented in Chapter 11. (a) Surface Brightness Profiles Fig. 2.19 shows the surface brightness profiles of three disk galaxies, as measured along their projected, major axes. A characteristic of these profiles is that they typically reveal a range over which µ (R) can be accurately fitted by a straight line. This corresponds to an exponential surface brightness profile L I(R) = I0 exp (−R/Rd ) , I0 = , (2.29) 2π R2d 50 Observational Facts Fig. 2.19. The surface brightness profiles of three disk galaxies plus their decomposition in an exponential disk (solid line) and a Sérsic bulge (dot-dashed line). [Based on data published in MacArthur et al. (2003) and kindly made available by L. MacArthur] Fig. 2.20. The effective radius (left panel) and the surface brightness at the effective radius (right panel) of disk dominated galaxies plotted against their absolute magnitude in the B-band. [Based on data published in Impey et al. (1996b)] (i.e. a Sérsic profile with n = 1). Here R is the cylindrical radius, Rd is the exponential scale- length, I0 is the central luminosity surface density, and L is the total luminosity. The effective radius enclosing half of the total luminosity is Re ≃ 1.67Rd. Following Freeman (1970) it has become customary to associate this exponential surface brightness profile with the actual disk component. The central regions of the majority of disk galaxies show an excess surface brightness with respect to a simple inward extrapolation of this exponential profile. This is interpreted as a contribution from the bulge component, and such interpretation is supported by images of edge- on disk galaxies, which typically reveal a central, roughly spheroidal, component clearly thicker than the disk itself (see e.g. NGC 4565 in Fig. 2.7). At large radii, the surface brightness profiles often break to a much steeper (roughly exponential) profile (an example is UGC 927, shown in Fig. 2.19). These breaks occur at radii Rb = α Rd with α in the range 2.5 to 4.5 (e.g. Pohlen et al., 2000; de Grijs et al., 2001). Fig. 2.20 shows Re and µe as functions of the absolute magnitude for a large sample of disk dominated galaxies (i.e. with a small or negligible bulge component). Clearly, as expected, more  2.3 Galaxies 51 luminous galaxies tend to be larger, although there is large scatter, indicating that galaxies of a given luminosity span a wide range in surface brightnesses. Note that, similar to ellipticals with MB > ∼ −20.5, more luminous disk galaxies on average have a higher surface brightness (see Fig. 2.14). When decomposing the surface brightness profiles of disk galaxies into the contributions of disk and bulge, one typically fits µ (R) with the sum of an exponential profile for the disk and a Sérsic profile for the bulge. We caution, however, that these bulge–disk decompositions are far from straightforward. Often the surface brightness profiles show clear deviations from a simple sum of an exponential plus Sérsic profile (e.g. UGC 12527 in Fig. 2.19). In addition, seeing tends to blur the central surface brightness distribution, which has to be corrected for, dust can cause significant extinction, and bars and spiral arms represent clear deviations from perfect axisymmetry. In addition, disks are often lop-sided (the centers of different isophotes are offset from each other in one particular direction) and can even be warped (the disk is not planar, but different disk radii are tilted with respect to each other). These difficulties can be partly overcome by using the full two-dimensional information in the image, by using color information to correct for dust, and by using kinematic information. Such studies require much detailed work and even then ambiguities remain. Despite these uncertainties, bulge–disk decompositions have been presented for large samples of disk galaxies (e.g. de Jong, 1996a; Graham, 2001; MacArthur et al., 2003). These studies have shown that more luminous bulges have a larger best-fit Sérsic index, similar to the relation found for elliptical galaxies (Fig. 2.13): while the relatively massive bulges of early-type spirals have surface brightness profiles with a best-fit Sérsic index n ∼ 4, the surface brightness profiles of bulges in late-type spirals are better fit with n < ∼ 1. In addition, the ratio between the effective radius of the bulge and the disk scale length is found to be roughly independent of Hubble type, with an average of ⟨re,b /Rd ⟩ = 0.22 ± 0.09. The fact that the bulge-to-disk ratio increases from late-type to early-type therefore indicates that brighter bulges have a higher surface brightness. Although the majority of bulges have isophotes that are close to elliptical, a non-negligible fraction of predominantly faint bulges in edge-on, late-type disk galaxies have isophotes that are extremely boxy, or sometimes even have the shape of a peanut. As we will see in §11.5.4, these peanut-shaped bulges are actually bars that have been thickened out of the disk plane. (b) Colors In general, disk galaxies are bluer than elliptical galaxies of the same luminosity. As discussed in §11.7, this is mainly owing to the fact that disk galaxies are still actively forming stars (young stellar populations are blue). Similar to elliptical galaxies, more luminous disks are redder, although the scatter in this color–magnitude relation is much larger than that for elliptical galaxies. Part of this scatter is simply due to inclination effects, with more inclined disks being more extincted and hence redder, although the intrinsic scatter (corrected for dust extinction) is still significantly larger than for ellipticals. In general, disk galaxies also reveal color gradients, with the outer regions being bluer than the inner regions (e.g. de Jong, 1996b). Although it is often considered standard lore that disks are blue and bulges are red, this is not supported by actual data. Rather, the colors of bulges are in general very similar to, or at least strongly correlated with, the central colors of their associated disks (e.g. de Jong, 1996a; Peletier & Balcells, 1996; MacArthur et al., 2004). Consequently, bulges also span a wide range in colors. (c) Disk Vertical Structure Galaxy disks are not infinitesimally thin. Observations suggest that the surface brightness distribution in the ‘vertical’ (z-) direction is largely independent of the distance R from the disk center. The three-dimensional luminosity density of the disk is therefore typically written in separable form as ν(R, z) = ν0 exp(−R/Rd ) f (z). (2.30) 52 Observational Facts A general fitting function commonly used to describe the luminosity density of disks in the z-direction is " # 2/n n|z| fn (z) = sech , (2.31) 2zd where n is a parameter controlling the shape of the profile near z = 0 and zd is called the scale height of the disk. Note that all these profiles project to face-on surface brightness profiles given by Eq. (2.29) with I0 = an ν0 zd , with an a constant. Three values of n have been used extensively in the literature: ⎧ ⎨ sech2 (z/2zd ) an = 4 n=1 fn (z) = sech(z/zd ) an = π n=2 (2.32) ⎩ exp(−|z|/zd ) an = 2 n = ∞. The sech2 -form for n = 1 corresponds to a self-gravitating isothermal sheet. Although this model has been used extensively in dynamical modeling of disk galaxies (see §11.1), it is generally recognized that the models with n = 2 and n = ∞ provide better fits to the observed surface brightness profiles. Note that all fn (z) decline exponentially at large |z|; they only differ near the mid-plane, where larger values of n result in steeper profiles. Unfortunately, since dust is usually concentrated near the mid-plane, it is difficult to accurately constrain n. The typical value of the ratio between the vertical and radial scale lengths is zd /Rd ∼ 0.1, albeit with considerable scatter. Finally, it is found that most (if not all) disks have excess surface brightness, at large distances from the mid-plane, that cannot be described by Eq. (2.31). This excess light is generally ascribed to a separate ‘thick disk’ component, whose scale height is typically a factor of 3 larger than for the ‘thin disk’. The radial scale lengths of thick disks, however, are remarkably similar to those of their corresponding thin disks, with typical ratios of Rd,thick /Rd,thin in the range 1.0–1.5, while the stellar mass ratios Md,thick /Md,thin decrease from ∼ 1 for low mass disks with Vrot < −1 ∼ 75 km s to ∼ 0.2 for massive disks with Vrot > −1 ∼ 150 km s (Yoachim & Dalcanton, 2006). (d) Stellar Halos The Milky Way contains a halo of old, metal-poor stars with a density dis- tribution that falls off as a power law, ρ ∝ r−α (α ∼ 3). In recent years, however, it has become clear that the stellar halo reveals a large amount of substructure in the form of stellar streams (e.g. Helmi et al., 1999; Yanny et al., 2003; Bell et al., 2008). These streams are associated with mater- ial that has been tidally stripped from satellite galaxies and globular clusters (see §12.2), and in some cases they can be unambiguously associated with their original stellar structure (e.g. Ibata et al., 1994; Odenkirchen et al., 2002). Similar streams have also been detected in our neighbor galaxy, M31 (Ferguson et al., 2002). However, the detection of stellar halos in more distant galaxies, where the individual stars cannot be resolved, has proven extremely difficult due to the extremely low surface brightnesses involved (typically much lower than that of the sky). Nevertheless, using extremely deep imaging, Sackett et al. (1994) detected a stellar halo around the edge-on spiral galaxy NGC 5907. Later and deeper observations of this galaxy suggest that this extraplanar emission is once again associated with a ring-like stream of stars (Zheng et al., 1999). By stacking the images of hundreds of edge- on disk galaxies, Zibetti et al. (2004) were able to obtain statistical evidence for stellar halos around these systems, suggesting that they are in fact rather common. On the other hand, recent observations of the nearby late-type spiral M33 seem to exclude the presence of a significant stellar halo in this galaxy (Ferguson et al., 2007). Currently the jury is still out as to what fraction of (disk) galaxies contain a stellar halo, and as to what fraction of the halo stars are associated with streams versus a smooth, spheroidal component. (e) Bars and Spiral Arms More than half of all spirals show bar-like structures in their inner regions. This fraction does not seem to depend significantly on the spiral type, and indeed S0  2.3 Galaxies 53 galaxies are also often barred. Bars generally have isophotes which are more squarish than ellipses and can be fit by the ‘generalized ellipse’ formula, (|x|/a)c + (|y|/b)c = 1, where a, b and c are constants and c is substantially larger than 2. Bars are, in general, quite elongated, with axis ratios in their equatorial planes ranging from about 2.5 to 5. Since it is difficult to observe bars in edge-on galaxies, their thickness is not well determined. However, since bars are so common, some limits may be obtained from the apparent thickness of the central regions of edge-on spirals. Such limits suggest that most bars are very flat, probably as flat as the disks themselves, but the bulges complicate this line of argument and it is possible that some bulges (for example, the peanut-shaped bulges) are directly related to bars (see §11.5.4). Galaxy disks show a variety of spiral structure. ‘Grand-design’ systems have arms (most fre- quently two) which can be traced over a wide range of radii and in many, but far from all, cases are clearly related to a strong bar or to an interacting neighbor. ‘Flocculent’ systems, on the other hand, contain many arm segments and have no obvious large-scale pattern. Spiral arms are clas- sified as leading or trailing according to the sense in which the spiral winds (moving from center to edge) relative to the rotation sense of the disk. Almost all spirals for which an unambiguous determination can be made are trailing. Spiral structure is less pronounced (though still present) in red light than in blue light. The spiral structure is also clearly present in density maps of atomic and molecular gas and in maps of dust obscuration. Since the blue light is dominated by massive and short-lived stars born in dense molecular clouds, while the red light is dominated by older stars which make up the bulk of the stellar mass of the disk, this suggests that spiral structure is not related to the star-formation process alone, but affects the structure of all components of disks, a conclusion which is more secure for grand-design than for flocculent spirals (see §11.6 for details). (f) Gas Content Unlike elliptical galaxies which contain gas predominantly in a hot and highly ionized state, the gas component in spiral galaxies is mainly in neutral hydrogen (HI) and molec- ular hydrogen (H2 ). Observations in the 21-cm lines of HI and in the mm-lines of CO have produced maps of the distribution of these components in many nearby spirals (e.g. Young & Scoville, 1991). The gas mass fraction increases from about 5% in massive, early-type spirals (Sa/SBa) to as much as 80% in low mass, low surface brightness disk galaxies (McGaugh & de Blok, 1997). In general, while the distribution of molecular gas typically traces that of the stars, the distribution of HI is much more extended and can often be traced to several Holmberg radii. Analysis of emission from HII regions in spirals provides the primary means for determining their metal abundance (in this case the abundance of interstellar gas rather than of stars). Metal- licity is found to decrease with radius. As a rule of thumb, the metal abundance decreases by an order of magnitude for a hundred-fold decrease in surface density. The mean metallicity also correlates with luminosity (or stellar mass), with the metal abundance increasing roughly as the square root of stellar mass (see §2.4.4). (g) Kinematics The stars and cold gas in galaxy disks move in the disk plane on roughly circular orbits. Therefore, the kinematics of a disk are largely specified by its rotation curve Vrot (R), which expresses the rotation velocity as a function of galactocentric distance. Disk rota- tion curves can be measured using a variety of techniques, most commonly optical long-slit or IFU spectroscopy of HII region emission lines, or radio or millimeter interferometry of line emis- sion from the cold gas. Since the HI gas is usually more extended than the ionized gas associated with HII regions, rotation curves can be probed out to larger galactocentric radii using spatially resolved 21-cm observations than using optical emission lines. Fig. 2.21 shows two examples of disk rotation curves. For massive galaxies these typically rise rapidly at small radii and then are almost constant over most of the disk. In dwarf and lower surface brightness systems a slower central rise is common. There is considerable variation from system to system, and features in rotation curves are often associated with disk structures such as bars or spiral arms. 54 Observational Facts Fig. 2.21. The rotation curves of the Sc galaxy NGC 3198 (left) and the low surface brightness galaxy F568- 3 (right). The curve in the left panel shows the contribution from the disk mass assuming a mass-to-light ratio of 3.8 M⊙ / L⊙. [Based on data published in Begeman (1989) and Swaters et al. (2000)] The rotation curve is a direct measure of the gravitational force within a disk. Assuming, for simplicity, spherical symmetry, the total enclosed mass within radius r can be estimated from 2 M(r) = rVrot (r)/G. (2.33) In the outer region, where Vrot (r) is roughly a constant, this implies that M(r) ∝ r, so that the enclosed mass of the galaxy (unlike its enclosed luminosity) does not appear to be converging. For the rotation curve of NGC 3198 shown in Fig. 2.21, the last measured point corresponds to an enclosed mass of 1.5 × 1011 M⊙ , about four times larger than the stellar mass. Clearly, the asymptotic total mass could even be much larger than this. The fact that the observed rotation curves of spiral galaxies are flat at the outskirts of their disks is evidence that they possess massive halos of unseen, dark matter. This is confirmed by studies of the kinematics of satellite galaxies and of gravitational lensing, both suggesting that the enclosed mass continues to increase roughly with radius out to at least 10 times the Holmberg radius. The kinematics of bulges are difficult to measure, mainly because of contamination by disk light. Nevertheless, the existing data suggests that the majority are rotating rapidly (consistent with their flattened shapes being due to the centrifugal forces), and in the same sense as their disk components. (h) Tully–Fisher Relation Although spiral galaxies show great diversity in luminosity, size, rotation velocity and rotation-curve shape, they obey a well-defined scaling relation between luminosity L and rotation velocity (usually taken as the maximum of the rotation curve well away from the center, Vmax ). This is known as the Tully–Fisher relation, an example of which is shown in Fig. 2.22. The observed Tully–Fisher relation is usually expressed in the form L = AVmax α , where A is the zero-point and α is the slope. The observed value of α is between 2.5 and 4, and is larger in redder bands (e.g. Pierce & Tully, 1992). For a fixed Vmax , the scatter in luminosity is typically 20%. This tight relation can be used to estimate the distances to spiral galaxies, using the principle described in §2.1.3(c). However, as we show in Chapter 11, the Tully–Fisher relation is also important for our understanding of galaxy formation and evolu- tion, as it defines a relation between dynamical mass (due to stars, gas, and dark matter) and luminosity.  2.3 Galaxies 55 –24 –22 – 5 log h –20 M –18 2 2.2 2.4 2.6 2.8 log W (km/s) Fig. 2.22. The Tully–Fisher relation in the I-band. Here W is the linewidth of the HI 21-cm line which is roughly equal to twice the maximum rotation velocity, Vmax. [Adapted from Giovanelli et al. (1997) by permission of AAS] 2.3.4 The Milky Way We know much more about our own Galaxy, the Milky Way, than about most other galaxies, simply because our position within it allows its stellar and gas content to be studied in consider- able detail. This ‘internal perspective’ also brings disadvantages, however. For example, it was not demonstrated until the 1920s and 30s that the relatively uniform brightness of the Milky Way observed around the sky does not imply that we are close to the center of the system, but rather is a consequence of obscuration of distant stars by dust. This complication, combined with the problem of measuring distances, is the main reason why many of the Milky Way’s large scale properties (e.g. its total luminosity, its radial structure, its rotation curve) are still substantially more uncertain than those of some external galaxies. Nevertheless, we believe that the Milky Way is a relatively normal spiral galaxy. Its main baryonic component is the thin stellar disk, with a mass of ∼ 5 × 1010 M⊙ , a radial scale length of ∼ 3.5 kpc, a vertical scale height of ∼ 0.3 kpc, and an overall diameter of ∼ 30 kpc. The Sun lies close to the mid-plane of the disk, about 8 kpc from the Galactic center, and rotates around the center of the Milky Way with a rotation velocity of ∼ 220 km s−1. In addition to this thin disk component, the Milky Way also contains a thick disk whose mass is 10–20% of that of the thin disk. The vertical scale height of the thick disk is ∼ 1 kpc, but its radial scale length is remarkably similar to that of the thin disk. The thick disk rotates slower than the thin disk, with a rotation velocity at the solar radius of ∼ 175 km s−1. In addition to the thin and thick disks, the Milky Way also contains a bulge component with a total mass of ∼ 1010 M⊙ and a half-light radius of ∼ 1 kpc, as well as a stellar halo, whose mass is only about 3% of that of the bulge despite its much larger radial extent. The stellar halo has a radial number density distribution n(r) ∝ r−α , with 2 < ∼α< ∼ 4, reaches out to at least 40 kpc, and shows no sign of rotation (i.e. its structure is supported against gravity by random rather than ordered motion). The structure and kinematics of the bulge are more complicated. The near-infrared image of the Milky Way, obtained with the COBE satellite, shows a modest, somewhat boxy bulge. As discussed in §11.5.4, it is believed that these boxy bulges are actually bars. This bar-like nature of the Milky Way bulge is supported by the kinematics of atomic and molecular gas in the inner few kiloparsecs (Binney et al., 1991), by microlensing measurements of the bu