The electromagnetic radiation field A
In this appendix, we will briefly review the most important The flux is measured in units of erg cm2 s1 Hz1.Ifthe properties of a radiation field. We thereby assume that the radiation field is isotropic, F vanishes. In this case, the same reader has encountered these quantities already in a different amount of radiation passes through the surface element in context. both directions. The mean specific intensity J is defined as the average of I over all angles, A.1 Parameters of the radiation field Z 1 J D d!I ; (A.3) The electromagnetic radiation field is described by the spe- 4 cific intensity I, which is defined as follows. Consider a D surface element of area dA. The radiation energy which so that, for an isotropic radiation field, I J.Thespecific passes through this area per time interval dt from within a energy density u is related to J according to solid angle element d! around a direction described by the n 4 unit vector , with frequency in the range between and u D J ; (A.4) C d,is c where u is the energy of the radiation field per D dE I dA cos dt d! d; (A.1) volume element and frequency interval, thus measured in erg cm3 Hz1. The total energy density of the radiation is where describes the angle between the direction n of the obtained by integrating u over frequency. In the same way, light and the normal vector of the surface element. Then, the intensity of the radiation is obtained by integrating the dA cos is the area projected in the direction of the infalling specific intensity I over . light. The specific intensity depends on the considered posi- tion (and, in time-dependent radiation fields, on time), the direction n, and the frequency . With the definition (A.1), A.2 Radiative transfer the dimension of I is energy per unit area, time, solid angle, and frequency, and it is typically measured in units of The specific intensity of radiation in the direction of propa- 2 1 1 1 erg cm s ster Hz . The specific intensity of a cosmic gation between source and observer is constant, as long as no source describes its surface brightness. emission or absorption processes are occurring. If s measures The specific net flux F passing through an area element the length along a line-of-sight, the above statement can be is obtained by integrating the specific intensity over all solid formulated as angles, dI Z D 0: (A.5) ds F D d!I cos : (A.2) An immediate consequence of this equation is that the sur- The flux that we receive from a cosmic source is defined in face brightness of a source is independent of its distance. The exactly the same way, except that cosmic sources usually observed flux of a source depends on its distance, because the subtend a very small solid angle on the sky. In calculating solid angle, under which the source is observed, decreases 2 the flux we receive from them, we may therefore drop the with the square of the distance, F / D [see (A.2)]. factor cos in (A.2); in this context, the specific flux is However, for light propagating through a medium, emission also denoted as S. However, in this Appendix (and only and absorption (or scattering of light) occurring along the here!), the notation S will be reserved for another quantity. path over which the light travels may change the specific
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 583 © Springer-Verlag Berlin Heidelberg 2015 584 A The electromagnetic radiation field intensity. These effects are described by the equation of the other hand, energy is added to the radiation field by 0 radiative transfer emission, accounted for by the -integral. Only a fraction 0 0 exp of this additional energy emitted at reaches dI the point , the rest is absorbed. D I C j : (A.6) ds In the context of (A.10), we call this a formal solution for the equation of radiative transport. The reason for this is The first term describes the absorption of radiation and states based on the fact that both the absorption and the emission that the radiation absorbed within a length interval ds is coefficient depend on the physical state of the matter through proportional to the incident radiation. The factor of propor- which radiation propagates, and in many situations this state tionality is the absorption coefficient , which has the unit of depends on the radiation field itself. For instance, and cm1.Theemission coefficient j describes the energy that j depend on the temperature of the matter, which in turn is added to the radiation field by emission processes, having depends, by heating and cooling processes, on the radiation a unit of erg cm3 s1 Hz1 ster1; hence, it is the radiation field it is exposed to. Hence, one needs to solve a coupled energy emitted per volume element, time interval, frequency system of equations in general: on the one hand the equation interval, and solid angle. Both, and j depend on the of radiative transport, and on the other hand the equation of nature and state (such as temperature, chemical composition) state for matter. In many situations, very complex problems of the medium through which light propagates. arise from this, but we will not consider them further in the The absorption and emission coefficients both account context of this book. for true absorption and emission processes, as well as the scattering of radiation. Indeed, the scattering of a photon can be considered as an absorption that is immediately followed by an emission of a photon. A.3 Blackbody radiation The optical depth along a line-of-sight is defined as the integral over the absorption coefficient, For matter in thermal equilibrium, the source function S is Z solely a function of the matter temperature, s 0 0 .s/ D ds .s /; (A.7) S D B.T / ; or j D B.T / ; (A.11) s0 where s0 denotes a reference point on the sightline from independent of the composition of the medium (Kirchhoff’s which the optical depth is measured. Dividing (A.6)by law). We will now consider radiation propagating through and using the relation d D ds in order to introduce matter in thermal equilibrium at constant temperature T . the optical depth as a new variable along the light ray, the Since in this case S D B .T / is constant, the solution equation of radiative transfer can be written as (A.10) can be written in the form
dI D DI C S ; (A.8) I./ I.0/ exp . / d Z C 0 0 B .T / d exp (A.12) where the source function 0 D C j I.0/ exp . / B.T / Œ1 exp . / : S D (A.9) From this it follows that I D B.T / is valid for sufficiently is defined as the ratio of the emission and absorption coeffi- large optical depth . The radiation propagating through cients. In this form, the equation of radiative transport can be matter which is in thermal equilibrium is described by the formally solved; as can easily be tested by substitution, the function B .T / if the optical depth is sufficiently large, solution is independent of the composition of the matter. A specific case of this situation can be illustrated by imagining the I./ D I.0/ exp ./ radiation field inside a box whose opaque walls are kept at a Z constant temperature T . Due to the opaqueness of the walls, C 0 0 0 d exp S./: (A.10) their optical depth is infinite, hence the radiation field within 0 the box is given by I D B.T /. This is also valid if the This equation has a simple interpretation. If I.0/ is the volume is filled with matter, as long as the latter is in thermal incident intensity, it will have decreased by absorption to equilibrium at temperature T . For these reasons, this kind of avalueI.0/ exp ./ after an optical depth of .On radiation field is also called blackbody radiation. A.3 Blackbody radiation 585
The function B.T / was first obtained in 1900 by Max Planck, and in his honor, it was named the Planck function; it reads 3 2hP 1 B.T / D ; (A.13) c2 ehP=kBT 1 27 where hP D 6:625 10 erg s is the Planck constant 16 1 and kB D 1:38 10 erg K is the Boltzmann constant. The shape of the spectrum (Fig. A.1) can be derived from statistical physics. Blackbody radiation is defined by I D B.T /,andthermal radiation by S D B .T /.Forlarge optical depths in the case of thermal radiation, the specific intensity converges to blackbody radiation. For small optical depth, the radiation field is approximated by an integral over the emissivity j, which can deviate strongly from that of blackbody spectrum even in the case of a thermal source; an example is the optically thin bremsstrahlung from the hot gas in galaxy clusters (see Sect. 6.4). The Planck function has its maximum at h P max 2:82 ; (A.14) kBT i.e., the frequency of the maximum is proportional to the temperature. This property is called Wien’s law. This law can also be written in more convenient units,
T D 5:88 1010 Hz : (A.15) max 1K
The Planck function can also be formulated depending on wavelength D c=, such that B.T / d D B.T / d, Fig. A.1 The Planck function (A.13) for different temperatures T .The 2 5 plot shows B .T / as a function of frequency , where high frequencies 2hPc = B.T / D : (A.16) are plotted towards the left (thus large wavelengths towards the right). exp .hPc=kBT/ 1 The exponentially decreasing Wien part of the spectrum is visible on the left, the Rayleigh–Jeans part on the right.Theshape of the spectrum Two limiting cases of the Planck function are of particular in the Rayleigh–Jeans part is independent of the temperature, which is interest. For low frequencies, h k T , one can apply the determining the amplitude however. Credit: T. Kaempf & M. Altmann, P B Argelander-Institut für Astronomie, Universität Bonn expansion of the exponential function for small arguments in (A.13). The leading-order term in this expansion then yields The energy density of blackbody radiation depends only 22 RJ D on the temperature, of course, and is calculated by integration B.T / B .T / kBT; (A.17) c2 over the Planck function, which is called the Rayleigh–Jeans approximation of the Z 1 4 4 Planck function. We point out that the Rayleigh–Jeans equa- u D dB.T / D B.T / D aT4 ; (A.19) c c tion does not contain the Planck constant, and this law had 0 been known even before Planck derived his exact equation. In where we defined the frequency-integrated Planck function the other limiting case of very high frequencies, h k T , P B Z the exponential factor in the denominator in (A.13) becomes 1 ac D D 4 very much larger than unity, so that we obtain B.T / dB.T / T ; (A.20) 0 4
3 2hP and where the constant a has the value B .T / BW.T / D e hP=kBT ; (A.18) c2 85k4 a D B D 7:56 1015 erg cm3 K4 : (A.21) called the Wien approximation of the Planck function. 3 3 15c hP 586 A The electromagnetic radiation field
The flux which is emitted by the surface of a blackbody per definition is chosen so as to yield the best agreement of the unit area is given by magnitude system with the visually determined magnitudes. j jD Z 1 Z 1 A difference of m 1 in this system corresponds to a 4 flux ratio of 2:51, and a flux ratio of a factor 10 or 100 F D dF D dB.T / D B.T / D SBT ; 0 0 corresponds to 2.5 or 5 magnitudes, respectively. (A.22) where the Stefan–Boltzmann constant SB has a value of A.4.2 Filters and colors
ac 25k4 D D B D 5:67 105 erg cm2 K4 s1 : Since optical observations are performed using a combina- SB 2 3 4 15c hP tion of a filter and a detector system, and since the flux ratios (A.23) depend, in general, on the choice of the filter (because the spectral energy distribution of the sources may be different), apparent magnitudes are defined for each of these filters. A.4 The magnitude scale The most common filters are shown in Fig. A.2 and listed in Table A.1, together with their characteristic wavelengths Optical astronomy was being conducted well before meth- and the widths of their transmission curves. The apparent ods of quantitative measurements became available. The magnitude for a filter X is defined as mX , frequently written Á brightness of stars had been cataloged more than 2000 years as X. Hence, for the B-band filter, mB B. ago, and their observation goes back as far as the ancient Next, we need to specify how the magnitudes measured world. Stars were classified into magnitudes, assigning a in different filters are related to each other, in order to define magnitude of 1 to the brightest stars and higher magnitudes to the color indices of sources. For this purpose, a particular the fainter ones. Since the apparent magnitude as perceived class of stars is used, main-sequence stars of spectral type by the human eye scales roughly logarithmically with the A0, of which the star Vega is an archetype. For such a star, radiation flux (which is also the case for our hearing), the by definition, U D B D V D R D I D :::, i.e., every magnitude scale represents a logarithmic flux scale. To link color index for such a star is defined to be zero. these visually determined magnitudes in historical catalogs For a more precise definition, let TX ./ be the trans- to a quantitative measure, the magnitude system has been mission curve of the filter-detector system. TX ./ specifies retained in optical astronomy, although with a precise defini- which fraction of the incoming photons with frequency tion. Since no historical astronomical observations have been are registered by the detector. The apparent magnitude of a conducted in other wavelength ranges, because these are not source with spectral flux S is then accessible to the unaided eye, only optical astronomy has to ÂR Ã RdTX ./ S bear the historical burden of the magnitude system. mX D2:5 log C const:; (A.25) dTX ./
A.4.1 Apparent magnitude where the constant needs to be determined from reference stars. We start with a relative system of flux measurements by Another commonly used definition of magnitudes is the considering two sources with fluxes S1 and S2.Theapparent AB system. In contrast to the Vega magnitudes, no stellar magnitudes of the two sources, m1 and m2,thenbehave spectral energy distribution is used as a reference here, but ref D according to instead one with a constant flux at all frequencies, S AB D 21 1 2 1 Â Ã S 2:89 10 erg s cm Hz . This value has S S been chosen such that A0 stars like Vega have the same 1 1 0:4.m1m2/ m1 m2 D2:5 log I D 10 : S2 S2 magnitude in the original Johnson-V-band as they have in AB D (A.24) the AB system, mV mV . With (A.25), one obtains for the conversion between the two systems This means that the brighter source has a smaller apparent WD AB Vega magnitude than the fainter one: the larger the apparent mAB!Vega mX mX magnitude, the fainter the source.1 The factor of 2.5 in this R ! dT ./ S AB D R X 2:5 log Vega (A.26) 1Of course, this convention is confusing, particularly to someone just dTX ./ S becoming familiar with astronomy, and it frequently causes confusion and errors, as well as problems in the communication with non- For the filters at the ESO Wide-Field Imager, which are astronomers—but we have to get along with that. designed to resemble the Johnson set of filters, the following A.4 The magnitude scale 587
Fig. A.2 Transmission curves of various filter-detector systems. From top to bottom: the filters of the NICMOS camera and the WFPC2 on-board HST, the Washington filter system, the filters of the EMMI instrument at ESO’s NTT, the filters of the WFI at the ESO/MPG 2.2-m telescope and those of the SOFI instrument at the NTT, and the Johnson-Cousins filters. In the bottom diagram, the spectra of three stars with different effective temperatures are displayed. Source: L. Girardi et al. 2002, Theoretical isochrones in several photometric systems. I. Johnson-Cousins-Glass, HST/WFPC2, HST/NICMOS, Washington, and ESO Imaging Survey filter sets, A&A 391, 195, p. 204, Fig. 3. c ESO. Reproduced with permission
Table A.1 For some of the best-established filter systems—Johnson, A.4.3 Absolute magnitude Strömgren, and the filters of the Sloan Digital Sky Surveys—the central (more precisely, the effective) wavelengths and the widths of the filters are listed The apparent magnitude of a source does not in itself tell us anything about its luminosity, since for the determination of Johnson UBVRI J H K L M the latter we also need to know its distance D in addition eff.nm/ 367 436 545 638 797 1220 1630 2190 3450 4750 to the radiative flux. Let L be the specific luminosity of .nm/ 66 94 85 160 149 213 307 39 472 460 a source, i.e., the energy emitted per unit time and per unit frequency interval, then the flux is given by (note that from
Strömgren uvbyˇw ˇn here on we switch back to the notation where S denotes the
eff.nm/ 349 411 467 547 489 489 flux, which was denoted by F earlier in this appendix) .nm/ 30191823153 L S D ; (A.27) 4D2 SDSS u0 g0 r0 i0 z0 eff.nm/ 354 477 623 762 913 where we implicitly assumed that the source emits isotrop- .nm/ 57 139 138 152 95 ically. Having the apparent magnitude as a measure of S (at the frequency defined by the filter which is applied), it is desirable to have a similar measure for L, specifying prescriptions are then to be applied: UAB D UVega C 0:80; the physical properties of the source itself. For this purpose, BAB D BVega 0:11; VAB D VVega ; RAB D RVega C 0:19; the absolute magnitude is introduced, denoted as MX ,where IAB D IVega C 0:59. X refers to the filter under consideration. By definition, MX 588 A The electromagnetic radiation field is equal to the apparent magnitude of a source if it were a distance modulus of D31:47. With these values, the to be located at a distance of 10 pc from us. The absolute absolute bolometric magnitude of the Sun becomes magnitude of a source is thus independent of its distance, in contrast to the apparent magnitude. With (A.27)wefindfor Mˇbol D mˇbol D 4:74 ; (A.31) the relation of apparent to absolute magnitude  à so that (A.30) can be written as D mX MX D 5 log 5 Á ; (A.28)  à 1 pc L Mbol D 4:74 2:5 log ; (A.32) Lˇ where we have defined the distance modulus in the final step. Hence, the latter is a logarithmic measure of the and the luminosity of the Sun is then distance of a source: D 0 for D D 10 pc, D 10 for D D D D 1 kpc, and 25 for D 1 Mpc. The difference 33 1 Lˇ D 3:85 10 erg s : (A.33) between apparent and absolute magnitude is independent of the filter choice, and it equals the distance modulus if no extinction is present. In general, this difference is modified The direct relation between bolometric magnitude and lumi- by the wavelength- (and thus filter-)dependent extinction nosity of a source can hardly be exploited in practice, because coefficient—see Sect. 2.2.4. the apparent bolometric magnitude (or the flux S) of a source cannot be observed in most cases. For observations of a source from the ground, only a limited window of frequen- A.4.4 Bolometric parameters cies is accessible. Nevertheless, in these cases one also likes to quantify the total luminosity of a source. For sources for The total luminosity L of a source is the integral of the which the spectrum is assumed to be known, like for many specific luminosity L over all frequencies. Accordingly, the stars, the flux from observations at optical wavelengths can total flux S of a source is the frequency-integrated specific be extrapolated to larger and smaller wavelengths, and so flux S.Theapparent bolometric magnitude mbol is defined mbol can be estimated. For galaxies or AGNs, which have as a logarithmic measure of the total flux, a much broader spectral distribution and which show much more variation between the different objects, this is not mbol D2:5 log S C const:; (A.29) feasible. In these cases, the flux of a source in a particular frequency range is compared to the flux the Sun would have where here the constant is also determined from reference at the same distance and in the same spectral range. If MX is stars. Accordingly, the absolute bolometric magnitude is the absolute magnitude of a source measured in the filter X, defined by means of the distance modulus, as in (A.28). The the X-band luminosity of this source is defined as absolute bolometric magnitude depends on the bolometric 0:4.MX MˇX / luminosity L of a source via LX D 10 LˇX : (A.34)
Mbol D2:5 log L C const:: (A.30) Thus, when speaking of, say, the ‘blue luminosity of a galaxy’, this is to be understood as defined in (A.34). For The constant can be fixed, e.g., by using the parameters of the reference, the absolute magnitude of the Sun in optical filters Sun: its apparent bolometric magnitude is mˇbol D26:83, is MˇU D 5:55, MˇB D 5:45, MˇV D 4:78, MˇR D 4:41, and the distance of one Astronomical Unit corresponds to and MˇI D 4:07. Properties of stars B
In this appendix, we will summarize the most important there are blue stars which are considerably hotter than the properties of stars as they are required for understanding the Sun, and red stars that are very much cooler. The temperature contents of this book. Of course, this brief overview cannot of a star can be estimated from its color. From the flux replace the study of other textbooks in which the physics of ratio at two different wavelengths or, equivalently, from the stars is covered in much more detail. color index X Y Á mX mY in two filters X and Y, the temperature Tc is determined such that a blackbody at Tc would have the same color index. Tc is called the color B.1 The parameters of stars temperature of a star. If the spectrum of a star was a Planck spectrum, then the equality Tc D Teff would hold, but in To a good approximation, stars are gas spheres, in the cores general these two temperatures differ. of which light atomic nuclei are transformed into heavier ones (mainly hydrogen into helium) by thermonuclear pro- cesses, thereby producing energy. The external appearance B.2 Spectral class, luminosity class, and of a star is predominantly characterized by its radius R and the Hertzsprung–Russell diagram its characteristic temperature T . The properties and evolution of a star depend mainly on its mass M . The spectra of stars can be classified according to the atomic In a first approximation, the spectral energy distribution (and, in cool stars, also molecular) spectral lines that are of the emission from a star can be described by a blackbody present. Based on the line strengths and their ratios, the spectrum. This means that the specific intensity I is given Harvard sequence of stellar spectra was introduced. These by a Planck spectrum (A.13) in this approximation. The spectral classes follow a sequence that is denoted by the luminosity L of a star is the energy radiated per unit time. letters O, B, A, F, G, K, M; besides these, some other spectral If the spectrum of star was described by a Planck spectrum, classes exist that will not be mentioned here. The sequence the luminosity would depend on the temperature and on the corresponds to a sequence of color temperature of stars: radius according to O stars are particularly hot, around 50 000 K, M stars very much cooler with Tc 3500 K. For a finer classification, D 2 4 L 4R SB T ; (B.1) each spectral class is supplemented by a number between 0 and 9. An A1 star has a spectrum very similar to that of an where (A.22) was applied. However, the spectra of stars devi- A0 star, whereas an A5 star has as many features in common ate from that of a blackbody (see Fig. 3.33 and the bottom with an A0 star as with an F0 star. panel of Fig. A.2). One defines the effective temperature Teff Plotting the spectral type versus the absolute magnitude of a star as the temperature a blackbody of the same radius for those stars for which the distance and hence the absolute would need to have to emit the same luminosity as the star, magnitude can be determined, a striking distribution of stars thus becomes apparent in such a Hertzsprung–Russell diagram L (HRD). Instead of the spectral class, one may also plot T 4 Á : (B.2) SB eff 4R2 the color index of the stars, typically B V or V I . The resulting color-magnitude diagram (CMD) is essentially The luminosities of stars cover a huge range; the weakest are equivalent to an HRD, but is based solely on photometric a factor 104 times less luminous than the Sun, whereas the data. A different but very similar diagram plots the luminos- brightest emit 105 times as much energy per unit time as ity versus the effective temperature. the Sun. This big difference in luminosity is caused either by In Fig. B.1, a color-magnitude diagram is plotted, a variation in radius or by different temperatures. We know compiled from data observed by the HIPPARCOS satellite. from the colors of stars that they have different temperatures: Instead of filling the two-dimensional parameter space
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 589 © Springer-Verlag Berlin Heidelberg 2015 590 B Properties of stars
Fig. B.1 Color-magnitude diagram for 41 453 individual stars, whose Fig. B.2 Schematic color-magnitude diagram in which the spectral parallaxes were determined by the Hipparcos satellite with an accu- types and luminosity classes are indicated. Source: http://de.wikipedia. racy of better than 20 %. Since the stars shown here are subject to org unavoidable strong selection effects favoring nearby and luminous stars, the relative number density of stars is not representative of their true abundance. In particular, the lower main sequence is much more densely giant branch, with their much higher luminosities compared populated than is visible in this diagram. Credit: European Space to main-sequence stars of the same spectral class, have a Agency, Web page of the Hipparcos project much larger radius than the corresponding main-sequence stars. This size effect is also observed spectroscopically: the rather uniformly, characteristic regions exist in such color- gravitational acceleration on the surface of a star (surface magnitude diagrams in which nearly all stars are located. gravity) is GM Most stars can be found in a thin band called the main g D : (B.3) sequence. It extends from early spectral types (O, B) with R2 high luminosities (‘top left’) down to late spectral types We know from models of stellar atmospheres that the width (K, M) with low luminosities (‘bottom right’). Branching off of spectral lines depends on the gravitational acceleration on from this main sequence towards the ‘top right’ is the domain the star’s surface: the lower the surface gravity, the narrower of red giants, and below the main sequence, at early spectral the stellar absorption lines. Hence, a relation exists between types and very much lower luminosities than on the main the line width and the stellar radius. Since the radius of sequence itself, we have the domain of white dwarfs. The a star—for a fixed spectral type or effective temperature— fact that most stars are arranged along a one-dimensional specifies the luminosity, this luminosity can be derived from sequence—the main sequence—is probably one of the most the width of the lines. In order to calibrate this relation, stars important discoveries in astronomy, because it tells us that of known distance are required. the properties of stars are determined basically by a single Based on the width of spectral lines, stars are classi- parameter: their mass. fied into luminosity classes: stars of luminosity class I are Since stars exist which have, for the same spectral type called supergiants, those of luminosity class III are giants, and hence the same color temperature (and roughly the same main-sequence stars are denoted as dwarfs and belong to effective temperature), very different luminosities, we can luminosity class V; in addition, the classification can be deduce immediately that these stars have different radii, further broken down into bright giants (II), subgiants (IV), as can be read from (B.2). Therefore, stars on the red and subdwarfs (VI). Any star in the Hertzsprung–Russell B.3 Structure and evolution of stars 591 diagram can be assigned a luminosity class and a spectral for the energy transport depends on the temperature profile class (Fig. B.2). The Sun is a G2 star of luminosity class V. inside the star. The intervals in a star’s radius in which energy If the distance of a star, and thus its luminosity, is transport takes place via convection are called convection known, and if in addition its surface gravity can be derived zones. Since in convection zones stellar material is subject from the line width, we obtain the stellar mass from these to mixing, the chemical composition is homogeneous there. parameters. By doing so, it turns out that for main-sequence In particular, chemical elements produced by nuclear fusion stars the luminosity is a steep function of the stellar mass, are transported through the star by convection. approximately described by Stars begin their lives with a homogeneous chemical com-  à position, resulting from the composition of the molecular L M 3:5 cloud out of which they are formed. If their mass exceeds : (B.4) Lˇ Mˇ about 0:08Mˇ, the temperature and pressure in their core are sufficient to ignite the fusion of hydrogen into helium. Therefore, a main-sequence star of M D 10Mˇ is 3000 Gassphereswithamassbelow 0:08Mˇ will not satisfy times more luminous than our Sun. these conditions, hence these objects—they are called brown dwarfs—are not stars in a proper sense.2 At the onset of nuclear fusion, the star is located on the zero-age main B.3 Structure and evolution of stars sequence (ZAMS) in the HRD (see Fig. B.3). The energy production by fusion of hydrogen into helium alters the To a very good approximation, stars are spherically sym- chemical composition in the stellar interior; the abundance metric. Therefore, the structure of a star is described by the of hydrogen decreases by the same rate as the abundance radial profile of the parameters of its stellar plasma. These of helium increases. As a consequence, the duration of this are density, pressure, temperature, and chemical composition phase of central hydrogen burning is limited. As a rough of the matter. During almost the full lifetime of a star, the estimate, the conditions in a star will change noticeably when plasma is in hydrostatic equilibrium, so that pressure forces about 10 % of its hydrogen is used up. Based on this criterion, and gravitational forces are of equal magnitude and directed the lifetime of a star on the main sequence can now be in opposite directions, so as to balance each other. estimated. The total energy produced in this phase can be The density and temperature are sufficiently high in the written as center of a star that thermonuclear reactions are ignited. In 2 EMS D 0:1 Mc 0:007 ; (B.5) main-sequence stars, hydrogen is fused into helium, thus four protons are combined into one 4He nucleus. For every helium where Mc2 is the rest-mass energy of the star, of which a nucleus that is produced this way, 26:73 MeV of energy fraction of 0.1 is fused into helium, which is supposed to are released. Part of this energy is emitted in the form of occur with an efficiency of 0.007. Phrased differently, in the neutrinos which can escape unobstructed from the star due fusion of four protons into one helium nucleus, an energy 2 to their very low cross section.1 The energy production rate of 0:007 4mpc is generated, with mp denoting the is approximately proportional to T 4 for temperatures below proton mass. In particular, (B.5) states that the total energy about 15 106 K, at which the reaction follows the so-called produced during this main-sequence phase is proportional to pp-chain. At higher temperatures, another reaction chain the mass of the star. In addition, we know from (B.4)that starts to contribute, the so-called CNO cycle, with an energy the luminosity is a steep function of the stellar mass. The production rate which is much more strongly dependent on lifetime of a star on the main sequence can then be estimated 20 temperature—roughly proportional to T . by equating the available energy EMS with the product of The energy generated in the interior of a star is trans- luminosity and lifetime. This yields ported outwards, where it is then released in the form of  à 2:5 electromagnetic radiation. This energy transport may take EMS 9 M=Mˇ 9 M tMSD 810 yr 8 10 yr : place in two different ways: first, by radiation transport, and L L=Lˇ Mˇ second, it can be transported by macroscopic flows of the (B.6) stellar plasma. This second mechanism of energy transport is called convection; here, hot elements of the gas rise upwards, Using this argument, we observe that stars of higher mass driven by buoyancy, and at the same time cool ones sink conclude their lives on the main sequence much faster than downwards. The process is similar to that observed in heating 2 water on a stove. Which of the two processes is responsible If the mass of a brown dwarf exceeds 0:013Mˇ, the central density and temperature are high enough to enable the fusion of deuterium (heavy hydrogen) into helium. However, the abundance of deuterium 1The detection of neutrinos from the Sun in terrestrial detectors was the is smaller by several orders of magnitude than that of normal hydrogen, final proof for the energy production mechanism being nuclear fusion. rendering the fuel reservoir of a brown dwarf very small. 592 B Properties of stars
Fig. B.3 Theoretical temperature-luminosity diagram of stars. The solid curve is the zero age main sequence (ZAMS), on which stars ignite the burning of hydrogen in their cores. The evolutionary tracks of these stars are indicated by the various lines which are labeled with the stellar mass. The hatched areas mark phases in which the evolution proceeds only slowly, so that many stars are observed to be in these areas. Source: A. Maeder & G. Meynet 1989, Grids of evolutionary models from 0.85 to 120 solar masses - Observational tests and the mass limits,A&A 210, 155, p. 166, Fig. 15. c ESO. Reproduced with permission
stars of lower mass. The Sun will remain on the main of hydrogen into helium takes place in a shell outside the sequence for about eight to ten billion years, with about center of the star. During this phase, the star quickly moves half of this time being over already. In comparison, very to the ‘right’ in the HRD, towards lower temperatures, and luminous stars, like O and B stars, will have a lifetime on thereby expands strongly. After this phase, the density and the main sequence of only a few million years before they temperature in the center rise so much as to ignite the fusion have exhausted their hydrogen fuel. of helium into carbon. A central helium-burning zone will In the course of their evolution on the main sequence, then establish itself, in addition to the source in the shell stars move away only slightly from the ZAMS in the HRD, where hydrogen is burned. As soon as the helium in the towards somewhat higher luminosities and lower effective core has been exhausted, a second shell source will form temperatures. In addition, the massive stars in particular can fusing helium. In this stage, the star will become a red giant lose part of their initial mass by stellar winds. The evolution or supergiant, ejecting part of its mass into the interstellar after the main-sequence phase depends on the stellar mass. medium in the form of stellar winds. Its subsequent evolu- Stars of very low mass, M . 0:7Mˇ, have a lifetime tionary path depends on this mass loss. A star with an initial on the main sequence which is longer than the age of the mass M . 8Mˇ will evolve into a white dwarf, which will Universe, therefore they cannot have moved away from the be discussed further below. main sequence yet. For stars with initial mass M . 2:5Mˇ, the helium burn- For massive stars, M & 2:5Mˇ, central hydrogen burning ing in the core occurs explosively, in a so-called helium flash. is first followed by a relatively brief phase in which the fusion A large fraction of the stellar mass is ejected in the course of B.3 Structure and evolution of stars 593 this flash, after which a new stable equilibrium configuration of about 0:6Mˇ and a radius roughly corresponding to that is established, with a helium shell source burning beside the of the Earth. hydrogen-burning shell. Expanding its radius, the star will If the initial mass of the star is &8Mˇ, the temperature evolve into a red giant or supergiant and move along the and density at its center become so large that carbon can asymptotic giant branch (AGB) in the HRD. also be fused. Subsequent stellar evolution towards a core- The configuration in the helium shell source is unstable, collapse supernova is described in Sect. 2.3.2. so that its burning will occur in the form of pulses. After The individual phases of stellar evolution have very differ- some time, this will lead to the ejection of the outer envelope ent time-scales. As a consequence, stars pass through certain which then becomes visible as a planetary nebula.The regions in the HRD very quickly, and for this reason stars at remaining central star moves to the left in the HRD, i.e., its those evolutionary stages are never or only rarely found in temperature rises considerably (to more than 105 K). Finally, the HRD. By contrast, long-lasting evolutionary stages like its radius gets smaller by several orders of magnitude, so the main sequence or the red giant branch exist, with those that the stars move downwards in the HRD, thereby slightly regions in an observed HRD being populated by numerous reducing its temperature: a white dwarf is born, with a mass stars. Units and constants C
In this book, we consistently used, besides astronomical X-ray astronomers measure energies in electron Volts, units, the Gaussian cgs system of units, with lengths mea- where 1eV D 1:602 1012 erg. Temperatures can also be sured in cm, masses in g, and energies in erg. This is the measured in units of energy, because kBT has the dimension commonly used system of units in astronomy. In these units, of energy. They are related according to 1 eV D 1:161 10 1 4 the speed of light is c D 2:998 10 cm s , the masses of 10 kB K. Since we always use the Boltzmann constant kB 24 protons, neutrons, and electrons are mp D 1:673 10 g, in combination with a temperature, its actual value is never 24 28 mn D 1:675 10 g, and me D 9:109 10 g, needed. The same holds for Newton’s constant of gravity respectively. which is always used in combination with a mass. Here one Frequently used units of length in astronomy include the has GMˇ Astronomical Unit, thus the average separation between the D 1:495 105 cm ; (C.1) Earth and the Sun, where 1 AU D 1:496 1013 cm, and the c2 parsec (see Sect. 2.2.1 for the definition), 1 pc D 3:086 which can also be written in the form 18 D 7 10 cm. A year has 1 yr 3:156 10 s. In addition, Â Ã D 2 masses are typically specified in Solar masses, 1Mˇ 3 pc km 33 G D 4:35 10 : (C.2) 1:989 10 g, and the bolometric luminosity of the Sun is Mˇ s 33 1 Lˇ D 3:846 10 erg s . In cgs units, the value of the elementary charge is e D The frequency of a photon is linked to its energy accord- 10 3=2 1=2 1 D 1 D 4:803 10 cm g s , and the unit of the magnetic ing to hP E, and we have the relation 1eV hP field strength is one Gauss, where 1 G D 1 g1=2 cm1=2 s1 D 2:418 1014 s1 D 2:418 1014 Hz. Accordingly, we can 1=2 3=2 1 erg cm . One of the very convenient properties of cgs write the wavelength D c= D hPc=E in the form units is that the energy density of the magnetic field in these D 2 h c units is given by B B =.8/—the reader may check that P D 1:2400 104 cm D 12 400 Å : the units of this equation are consistent. 1 eV
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 595 © Springer-Verlag Berlin Heidelberg 2015 Recommended literature D
In the following, we will give some recommendations for further study of the literature on astrophysics. For readers D.2 More specific literature who have been in touch with astronomy only occasionally until now, the general textbooks may be of particular interest. More specific monographs and textbooks exist for the indi- The choice of literature presented here is a very subjective vidual topics covered in this book, some of which shall be one which represents the preferences of the author, and suggested below. Again, this is just a brief selection. The of course it represents only a small selection of the many technical level varies substantially among these books and, astronomy texts available. in general, exceeds that of the present text.
Astrophysical processes: • M. Harwit: Astrophysical Concepts, Springer, New-York, D.1 General textbooks 2006, • G.B. Rybicki & A.P. Lightman: Radiative Processes in There exist a large selection of general textbooks in astron- Astrophysics, John Wiley & Sons, New York, 1979, omy which present an overview of the field at a non-technical • F. Shu: The Physics of Astrophysics I: Radiation, level. A classic one (though by now becoming of age) and an University Science Books, Mill Valley, 1991, excellent presentation of astronomy is • F. Shu: The Physics of Astrophysics II: Gas Dynamics, • F. Shu: The Physical Universe: An Introduction to Astron- University Science Books, Mill Valley, 1991, omy, University Science Books, Sausalito, 1982. • S.N. Shore: The Tapestry of Modern Astrophysics, Wiley- Turning to more technical books, at about the level of the VCH, Berlin, 2002, present text, my favorite is • D.E. Osterbrock & G.J. Ferland: Astrophysics of Gaseous • B.W. Carroll & D.A. Ostlie: An Introduction to Modern Nebulae and Active Galactic Nuclei, University Science Astrophysics, Addison-Wesley, Reading, 2006; Books, Mill Valley, 2005. Furthermore, there is a three-volume set of books, its 1400 pages cover the whole range of astronomy. The • T. Padmanabhan: Theoretical Astrophysics: I. Astrophys- texts ical Processes. II. Stars and Stellar Systems. III. Galaxies • M.L. Kutner: Astronomy: A physical perspective, and Cosmology, Cambridge University Press, Cambridge, Cambridge University Press, Cambridge, 2003, 2000. • J.O. Bennett, M.O. Donahue, N. Schneider & M. Voit: The Cosmic Perspective, Addison-Wesley, 2013, Galaxies and gravitational lenses: also cover the whole field of astronomy. A text with a • L.S. Sparke & J.S. Gallagher: Galaxies in the Uni- particular focus on stellar and Galactic astronomy is verse: An Introduction, Cambridge University Press, • A. Unsöld & B. Baschek: The New Cosmos, Springer- Cambridge, 2007, Verlag, Berlin, 2002; • J. Binney & M. Merrifield: Galactic Astronomy, Princeton The book University Press, Princeton, 1998, • M.H. Jones & R.J.A. Lambourne: An Introduction to • J. Binney & S. Tremaine: Galactic dynamics, Princeton Galaxies and Cosmology, Cambridge University Press, University Press, Princeton, 2008, Cambridge, 2003 • R.C. Kennicutt, Jr., F. Schweizer & J.E. Barnes: Galax- covers the topics described in this book and is also highly ies: Interactions and Induced Star Formation, Saas-Fee recommended; it is less technical than the present text. Advanced Course 26, Springer-Verlag, Berlin, 1998,
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 597 © Springer-Verlag Berlin Heidelberg 2015 598 D Recommended literature
• B.E.J. Pagel: Nucleosynthesis and Chemical Evolution of only. In Physics Reports (Phys. Rep.) and Reviews of Galaxies, Cambridge University Press, Cambridge, 2009, Modern Physics (RMP), astronomical review articles are • F.Combes,P.Boissé,A.Mazure&A.Blanchard:Galax- also frequently found. Such articles are also published in ies and Cosmology, Springer-Verlag, 2004, the lecture notes of international summer/winter schools and • P. Schneider, J. Ehlers & E.E. Falco: Gravitational Lenses, in the proceedings of conferences; of particular note are Springer-Verlag, New York, 1992. the Lecture Notes of the Saas-Fee Advanced Courses.A • P. Schneider, C.S. Kochanek & J. Wambsganss: very useful archive containing review articles on the topics Gravitational Lensing: Strong, Weak & Micro, Saas- covered in this book is the Knowledgebase for Extragalactic Fee Advanced Course 33, G. Meylan, P. Jetzer & P. North Astronomy and Cosmology, which can be found at (Eds.), Springer-Verlag, Berlin, 2006. http://nedwww.ipac.caltech.edu/level5. Original astronomical research articles are published in Active galaxies: the relevant scientific journals; most of the figures pre- • B.M. Peterson: An Introduction to Active Galactic Nuclei, sented in this book are taken from these journals. The Cambridge University Press, Cambridge, 1997, most important of them are Astronomy & Astrophysics • R.D. Blandford, H. Netzer & L. Woltjer: Active Galactic (A&A), The Astronomical Journal (AJ), The Astrophysical Nuclei, Saas-Fee Advanced Course 20, Springer-Verlag, Journal (ApJ), Monthly Notices of the Royal Astronomical 1990, Society (MNRAS), and Publications of the Astronomical • J. Krolik: Active Galactic Nuclei, Princeton University Society of the Pacific (PASP). Besides these, a number of Press, Princeton, 1999, smaller, regional, or more specialized journals exist, such as • J. Frank, A. King & D. Raine: Accretion Power in Astro- Astronomische Nachrichten (AN), Acta Astronomica (AcA), physics, Cambridge University Press, Cambridge, 2012. or Publications of the Astronomical Society of Japan (PASJ). Some astronomical articles are also published in the journals Cosmology: Nature and Science. The Physical Review D and Physical • M.S. Longair: Galaxy Formation, Springer-Verlag, Review Letters contain an increasing number of papers on Berlin, 2008, astrophysical cosmology. • J.A. Peacock: Cosmological Physics, Cambridge Since many years now, the primary source of astronomical University Press, Cambridge, 1999, information by far is the electronic archive • T. Padmanabhan: Structure formation in the Universe, http://arxiv.org/archive/astro-ph Cambridge University Press, Cambridge, 1993, which is freely accessible. This archive, now hosted at • E.W. Kolb and M.S. Turner: The Early Universe, Addison Cornell University and supported by the Simons Foundation Wesley, 1990, and the Allianz der deutschen Wissenschaftsorganisationen, • S. Dodelson: Modern Cosmology, Academic Press, San koordiniert durch TIB, MPG und HGF, has existed since Diego, 2003, 1992, with an increasing number of articles being stored • P.J.E. Peebles: Principles of Physical Cosmology, Prince- at this location. In particular, in the fields of extragalactic ton University Press, Princeton, 1993, astronomy and cosmology, almost all articles that are pub- • G. Börner: The Early Universe, Springer-Verlag, Berlin, lished in the major journals can be found in this archive. A 2003, large number of review articles and coference proceedings • D.H. Lyth & A.R. Liddle: The Primordial Density Pertur- are also available here. bation: Cosmology, Inflation and the Origin of Structure, The SAO/NASA Astrophysics Data System (ADS) is a Cambridge University Press, Cambridge, 2009. Digital Library portal for Astronomy and Physics, operated by the Smithsonian Astrophysical Observatory (SAO) under a NASA grant. It can be accessed via the Internet at, e.g., D.3 Review articles, current literature, http://cdsads.u-strasbg.fr/abstract_service.html, and journals http://adsabs.harvard.edu/abstract_service.html, and it provides the best access to astronomical literature. Besides textbooks and monographs, review articles on Besides tools to search for authors and keywords, ADS offers specific topics are particularly useful for getting extended also direct access to older articles that have been scanned. information about a special field. A number of journals The access to more recent articles, and to all articles in some and series exist in which excellent review articles are other journals, is restricted to IP addresses that are associated published. Among these are Annual Reviews of Astronomy with a subscription for the respective journals—but ADS also and Astrophysics (ARA&A) and Astronomy & Astrophysics contains a link to the article in the arXiv (if it has been posted Reviews (A&AR), both publishing astronomical articles there), so also these articles are accessible. Acronyms used E
In this Appendix, we compile some of the acronyms that are BOOMERANG – Balloon Observations Of Millimetric used, and references to the sections in which these acronyms Extragalactic Radiation and Geophysics have been introduces or explained. (Sect. 8.6.4) 2dF(GRS) – 2 degree Field Galaxy Redshift Survey BTP diagram – Baldwin–Phillips–Terlevich diagram (Sect. 8.1.2) (Sect. 5.4.3) 2MASS – Two Micron All Sky Survey (Sect. 1.4) CBI – Cosmic Background Imager (Sect. 8.6.5) AAS – American Astronomical Society CCAT – Cerro Chajnantor Atacama Telescope AAT – Anglo-Australian Telescope (Sect. 1.3.3) (Chap. 11) ACBAR – Arcminute Cosmology Bolometer Array CCD – Charge Coupled Device – Receiver (Sect. 8.6.5) CDF – Chandra Deep Field (Sect. 9.2.1) ACO – Abell, Corwin & Olowin (catalogue of CDM – Cold Dark Matter (Sect. 7.4.1) clusters of galaxies, Sect. 6.2.1) CERN – Conseil European pour la Recherché ACS – Advanced Camera for Surveys (HST Nucleaire instrument—Sect. 1.3.3) CfA – Harvard-Smithsonian Center for ACT – Atacama Cosmology Telescope Astrophysics (Sect. 8.6.6) CFHT – Canada-France-Hawaii Telescope ADAF – Advection-Dominated Accretion Flow (Sect. 1.3.3) (Sect. 5.3.2) CFRS – Canada-France Redshift Survey AGB – Asymptotic Giant Branch (Sect. 3.5.2) (Sect. 8.1.2) AGN – Active Galactic Nucleus (Sect. 5) COSMOS – Cosmological Evolution Survey ALMA – Atacama Large Millimeter/sub-millimeter (Sect. 9.2.1) Array (Sect. 1.3.1) CTIO – Cerro Tololo Inter-American Observatory AMR – Adaptive Mesh Refinement (Sect. 10.6.1) CXB – Cosmic X-ray Background (Sect. 9.5.3) APEX – Atacama Pathfinder Experiment DASI – Degree Angular Scale Interferometer (Sect. 1.3.1) (Sect. 8.6.4) ASP – Astronomical Society of the Pacific DES – Dark Energy Survey (Chap. 11) AU – Astronomical Unit DIRBE – Diffuse Infrared Background Experiment BAL – Broad Absorption Line (-Quasar, (instrument onboard COBE) Sect. 5.7) DLA system – Damped Lyman Alpha system BAOs – Baryonic Acoustic Oscillations (Sect. 9.3.4) (Sect. 7.4.3) DRG – Distant Red Galaxy (Sect 9.1.3) BATSE – Burst And Transient Source Experiment dSph – dwarf Spheroidal (Sect. 3.2.1) (CGRO instrument, Sect. 9.7) DSS – Digital Sky Survey (Sect. 1.4) BBB – Big Blue Bump (Sect. 5.4.1) ECDFS – Extended Chandra Deep Field South BBN – Big Bang Nucleosynthesis (Sect. 4.4.5) (Sect. 9.3.3) BCD – Blue Compact Dwarf (Sect. 3.2.1) EdS – Einstein–de Sitter (Sect. 4.3.4) BCG – Brightest Cluster Galaxy (Sect. 6.2.4) E-ELT – European Extremely Large Telescope BH – Black Hole (Sect. 5.3.5) (Chap. 11) BLR – Broad Line Region (Sect. 5.4.2) EMSS – Extended Medium Sensitivity Survey BLRG – Broad Line Radio Galaxy (Sect. 5.2.4) (Sect. 6.4.5)
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 599 © Springer-Verlag Berlin Heidelberg 2015 600 EAcronymsused
EPIC – European Photon Imaging Camera HLS – Herschel Lensing Survey (Sect. 9.2.3) (XMM-Newton instrument) HRD – Hertzsprung–Russell Diagram (Appendix B) ERO – Extremely Red Object (Sect. 9.3.2) HRI – High Resolution Imager (ROSAT EROS – Expérience pour la Recherche d’Objets instrument) Sombres (microlensing collaboration, HST – Hubble Space Telescope (Sect. 1.3.3) Sect. 2.5) HVC – High Velocity Cloud (Sect. 2.3.6) ESA – European Space Agency HUDF – Hubble Ultradeep Survey (Sect. 9.2.1) ESO – European Southern Observatory IAU – International Astronomical Union FFT – Fast Fourier Transform (Sect. 7.5.3) ICL – IntraCluster Light (Sect. 6.3.4) FIR – Far Infrared ICM – Intra-Cluster Medium (Chap. 6) FIRAS – Far Infrared Absolute Spectrophotometer IFU – Integral Field Unit (Sect. 1.3.3) (instrument onboard COBE; see Fig. 4.3) IGM – Intergalactic Medium (Sect. 10.3) FJ – Faber–Jackson (Sect. 3.4.2) IMF – Initial Mass Function (Sect. 3.5.1) FOC – Faint Object Camera (HST instrument) IoA – Institute of Astronomy (Cambridge) FORS – Focal Reducer / Low Dispersion IR – Infrared (Sect. 1.3.2) Spectrograph (VLT instrument) IRAC – Infrared Array Camera (instrument on FOS – Faint Object Spectrograph (HST instrument) Spitzer—Sect. 1.3.2) FP – Fundamental Plane (Sect. 3.4.3) IRAS – Infrared Astronomical Observatory FR (I/II) – Fanaroff–Riley Type (Sect. 5.1.2) (Sect. 1.3.2) FUSE – Far Ultraviolet Spectroscopic Explorer IRS – Infrared Spectrograph (instrument on (Sect. 1.3.4) Spitzer—Sect. 1.3.2) FWHM – Full Width Half Maximum (Sect. 5.1.4) ISM – Interstellar Medium GALEX – Galaxy Evolution Explorer (Sect. 1.3.4) ISO – Infrared Space Observatory (Sect. 1.3.2) GBM – Gamma-ray Burst Monitor (instrument on ISW effect – Integrated Sachs–Wolfe effect (Sect. 8.6.1) Fermi—Sect. 1.3.6) IUE – International Ultraviolet Explorer GC – Galactic Center (Sects. 2.3, 2.6) (Sect. 1.3.4) GEMS – Galaxy Evolution from Morphology and IVC – Intermediate-Velocity Cloud (Sect. 2.3.7) Spectral Energy Distributions (Sect. 9.2.1) JCMT – James Clerk Maxwell Telescope (Sect. 1.3.1) GGL – Galaxy-Galaxy Lensing (Sect. 7.7) JVAS – Jodrell Bank-VLA Astrometric Survey GMT – Giant Magellan Telescope (Chap. 11) (Sect. 3.11.3) GOODS – Great Observatories Origins Deep Survey JWST – James Webb Space Telescope (Chap. 11) (Sect. 9.2.1) KAO – Kuiper Airborne Observatory (Sect. 1.3.2) GR – General Relativity KiDS – KiLO Degree Survey (Chap. 11) GRB – Gamma-Ray Burst (Sect. 1.3.5, 9.7) LAB – Leiden-Argentine-Bonn (Sect. 1.4) GTC – Gran Telescopio Canarias (Sect. 1.3.3) LAE – Lyman Alpha Emitter (Sect 9.1.3) GUT – Grand Unified Theory (Sect. 4.5.3) LAT – Large Area Telescope (instrument on Gyr – Gigayear D 109 years Fermi—Sect. 1.3.6) GZK – Greisen–Zatsepin–Kuzmin (Sect. 2.3.4) LBG – Lyman-Break Galaxy (Sect. 9.1.1) HB – Horizontal Branch LBT – Large Binocular Telescope (Sect. 1.3.3) HCG – Hickson Compact Group (catalogue of LCRS – Las Campanas Redshift Survey (Sect. 8.1.2) galaxy groups, Sect. 6.2.3) LFI – Low-Frequency Instrument (onboard the HDF(N/S) – Hubble Deep Field (North/South) Planck satellite; Sect. 8.6.6) (Sect. 1.3.3, 9.2.1) LHC – Large Hadron Collider HDM – Hot Dark Matter (Sect. 7.4.1) LINER – Low-Ionization Nuclear Emission-Line HEAO – High Energy Astrophysical Observatory Region (Sect. 5.2.3) (Sect. 1.3.5) LIRG – Luminous InfraRed Galaxy (Sect. 9.4.1) H.E.S.S. – High Energy Stereoscopic System LISA – Laser Interferometer Space Antenna (Sect. 1.3.6) (Chap. 11) HFI – High-Frequency Instrument (onboard the LMC – Large Magellanic Cloud Planck satellite; Sect. 8.6.6) LMT – Large Millimeter Telescope (Chap. 11) HIFI – Heterodyne Instrument for the Far Infrared LOFAR – Low Frequency Array (Chap. 11) (Herschel instrument—Sect. 1.3.2) LSB galaxy – Low Surface Brightness galaxy (Sect. 3.3.2) EAcronymsused 601
LSR – Local Standard of Rest (Sect. 2.4.1) QSO – Quasi-Stellar Object (Sect. 5.2.1) LSS – Large-Scale Structure (Chap. 8) RASS – ROSAT All-Sky Survey (Sect. 6.4.5) LSST – Large Synoptic Survey Telescope (Chap. 11) RCS – Red Cluster Sequence (Sect. 6.8) MACHO – Massive Compact Halo Object (and REFLEX – ROSAT-ESO Flux-Limited X-Ray survey collaboration of the same name, Sect. 2.5) RGB – Red Giant Branch (Sect. 3.5.2) MAGIC – Major Atmospheric Gamma-ray Imaging ROSAT – Roentgen Satellite (Sect. 1.3.5) Cherenkov telescope (Sect. 1.3.6) SAO – Smithsonian Astrophysical Observatory MAMBO – Max-Planck Millimeter Bolometer SCUBA – Sub-millimeter Common-User Bolometer (Sect. 9.3.3) Array (Sect. 1.3.1) MAXIMA – Millimeter Anisotropy Experiment Imaging SDSS – Sloan Digital Sky Survey (Sects. 1.4, 8.1.2) Array (Sect. 8.6.4) SFR – Star Formation Rate (Sect. 9.6.1) MDM – Mixed Dark Matter (Sect. 7.4.2) SGP – South Galactic Pole (Sect. 2.1) MIPS – Multiband Imaging Photometer for Spitzer SIS – Singular Isothermal Sphere (Sect. 3.11.2) (instrument on Spitzer— Sect. 1.3.2) SKA – Square Kilometer Array (Chap. 11) MIR – Mid-Infrared SLACS – Sloan Lens Advanced Camera for Surveys MLCS – Multi-Color Light Curve Shape (Sect. 3.9.4) (Sect. 3.11.3) MMT – Multi-Mirror Telescope SMBH – Supermassive Black Hole (Sect. 5.3) MOND – Modified Newtonian Dynamics (Chap. 11) SMC – Small Magellanic Cloud MS – used for the ‘Main Sequence’ of stars, or the SMG – Sub-Millimeter Galaxy (Sect. 9.3.3) ‘Millennium Simulation’ (Sect. 7.5.3) SN(e) – Supernova(e) (Sect. 2.3.2) MW – Milky Way SNR – Supernova Remnant MXXL – Millennium XXL simulation (Sect. 7.5.3) SOFIA – Stratospheric Observatory for Infrared NAOJ – National Astronomical Observatory of Japan Astronomy (Sect. 1.3.2) NFW – Navarro, Frenk & White (-profile, SPH – Smooth Particle Hydrodynamics Sect. 7.6.1) (Sect. 10.6.1) NGC – New General Catalog (Chap. 3) SPIRE – Spectral and Photometric Imaging REceiver NGP – North Galactic Pole (Sect. 2.1) (Herschel instrument— Sect. 1.3.2) NICMOS – Near Infrared Camera and Multi-Object SPT – South Pole Telescope (Sect. 1.3.1) Spectrometer (HST instrument— Sect. 1.3.3) SQLS – SDSS Quasar Lens Search (Sect. 3.11.3) NIR – Near Infrared STIS – Space Telescope Imaging Spectrograph NLR – Narrow Line Region (Sect. 5.4.3) (HST instrument) NLRG – Narrow Line Radio Galaxy (Sect. 5.2.4) STScI – Space Telescope Science Institute NOAO – National Optical Astronomy Observatory (Sect. 1.3.3) NRAO – National Radio Astronomy Observatory SZ – Sunyaev–Zeldovich (-effect, Sect. 6.4.4) NTT – New Technology Telescope (Sect. 1.3.3) TDE – Tidal Disruption Event (Sect. 5.5.6) NVSS – NRAO VLA Sky Survey (Sect. 1.4) TeVeS – Tensor-Vector-Scalar (Chap. 11) OGLE – Optical Gravitational Lensing Experiment TF – Tully–Fisher (Sect. 3.4) (microlensing collaboration, Sect. 2.5) TMT – Thirty Meter Telescope (Chap. 11) OVV – Optically Violently Variable (Sect. 5.2.5) TP-AGB – Thermally Pulsating AGB star (Sect. 3.5.5) PACS – Photodetector Array Camera and star Spectrometer (Herschel instrument— UDF – Ultra Deep Field (Sect. 9.2.1) Sect. 1.3.2) UHECRs – Ultra-High Energy Cosmic Rays PL – Period-Luminosity (Sect. 2.2.7) (Sect. 2.3.4) PLANET – Probing Lensing Anomalies Network ULIRG – Ultraluminous Infrared Galaxy (Sect. 9.3.1) (microlensing collaboration, Sect. 2.5) ULX – Ultraluminous Compact X-ray Source PM – Particle-Mesh (Sect. 7.5.3) (Sect. 9.3.1) P3M – Particle-Particle Particle-Mesh (Sect. 7.5.3) UV – Ultraviolet PN – Planetary Nebula VISTA – Visible and Infrared Survey Telescope POSS – Palomar Observatory Sky Survey (Sect. 1.4) (Sect. 1.3.3) PSF – Point Spread Function VLA – Very Large Array (Sect. 1.3.1) PSPC – Position-Sensitive Proportional Counter VLBA –VeryLongBaselineArray(Sect.1.3.1) (ROSAT instrument) VLBI – Very Long Baseline Interferometer QCD – Quantum Chromodynamics (Sect. 4.4.1) (Sect. 1.3.1) 602 EAcronymsused
VLT – Very Large Telescope (Sect. 1.3.3) WFI – Wide Field Imager (camera at the ESO/MPG VST – VLT Survey Telescope (Sect. 1.3.3) 2.2m telescope, La Silla, Sect. 6.6.2) VVDS – VIMOS VLT Deep Survey (Sect. 8.1.2) WFPC2 – Wide Field and Planetary Camera 2 (HST WD – White Dwarf (Sect. 2.3.2) instrument– Sect. 1.3.3) WDM – Warm Dark Matter (Sect. 7.8) WMAP – Wilkinson Microwave Anisotropy Probe WFIRST – Wide Field Infrared Space Telescope (Sect. 8.6.5) (Chap. 11) XDF – eXtremely Deep Field (Sect. 9.2.1) WIMP – Weakly Interacting Massive Particle XMM – X-ray Multi-Mirror Mission (Sect. 1.3.5) (Sect. 4.4.3) XRB – X-Ray Background (Sect. 9.5.3) WISE – Wide-field Infrared Survey ZAMS – Zero Age Main Sequence (Sect. 3.5.2) Explorer(Sect.1.3.2) WFC3 – Wide Field Camera 3 (HST instrument—Sect. 1.3.3) Solutions to problems F
Solution to 1.1. If the object at current distance D had a The fraction of the sky covered by these galaxies is f D 2 constant velocity v D H0D for all times, it needed a time !=.4/ D R n0 r0 0:6 %. t D D=v D 1=H0 to reach separation D. This time is independent of D. Using (1.7), we find that Solution to 1.3. (1) The mean baryon density of the Universe is b D 24 2 3:086 10 cm 0:15˝ 3H =.8G/. Making use of (1.14), this yields H 1 D D 9:77 h 1 109 ; m 0 0 7 s yr 31 3 h10 cm b D 4:3 10 gcm . The estimate to the local mass 3 33 density yields local D 1Mˇ pc 2 10 g .3 7 whereweusedthat1 yr D 3:16 10 s. For a value of h 1018 cm/3 D .2=27/1021 g=cm3 71023 g=cm3. 0:71, this time is comparable to, but slightly larger than the 8 Thus, local= b 1:6 10 . age of the oldest stars. Light can propagate a distance c=H0 (2) According to (1.1), the mass of the Galaxy inside R is 0 over the time-scale H 1,where D 2 0 M R0 V0 =G, yielding a mean density within R0 of 5 1 c 2:998 10 km s 1 2 D D 2:998h Gpc : M V0 1 1 D D H0 100h km s Mpc 8 3 2 : .4=3/ R0 .4=3/ G R0 Solution to 1.2. The number of galaxies in a sphere of D D 1 D 3 D The mean matter density of the Universe is m radius r0 1h Gpc is N .4=3/r0 n0,wheren0 ˝ 3H 2=.8G/ ˝ 0:3 2 10 2 h3 Mpc 3. Thus, m 0 . With m , this yields
V 2 8G 2V 2 N D .4=3/h 3 Gpc3 2 10 2 h3 Mpc 3 8 D 0 D 0 2 2 2 2 m .4=3/ G R ˝m 3H ˝m R H D 7 7 0 0 0 0 .8=3/ 10 8 10 : Â Ã 2 220 = 2 D km s 2 1 The number of these galaxies per square degree on the sky is h ˝m 8 kpc 100 .km=s/ Mpc obtained by dividing N by the solid angle of the sky, which ı 2 2 is 4 steradian. Since 180 corresponds to rad, we have D 0:275 103 1:0 106 : 2 2 2 that 1 steradian D .180=/ deg , so that the full sky has a h ˝m solid angle of 4 .180=/2 deg2 D 41253 deg2. This yields a number density of 2 103 deg2. (3) The mean number of baryons Nb in the box is the volume To calculate the fraction of the sky covered by the lumi- of the box times the mean number density of baryons. nous region of these galaxies, we consider first a thin spher- The latter is given by the mean mass density b of ical shell of radius r and thickness dr around us. In this baryons in the Universe, divided by the mass per baryon, 24 2 which is mb 1:7 10 g. Thus, making use of value shell, there are dN D 4r n0 dr galaxies, each of them subtending a solid angle of R2=r2,whereR D 10 kpc is the of b derived above, radius of the luminous region. Thus, the solid angle covered 2 2 3 31 3 24 1 by all galaxies in the shell is d! D 4 R n0 dr. The solid Nb 1 m 4:3 10 g=cm .1:7 10 g/ angle covered by all galaxies within distance r0 is obtained .0:43=1:7/ 0:25 : by integrating this expression over r, Z Z r0 r0 Thus, the mean baryon density in the Universe is about 2 2 2 2 3 ! D d! D 4 R n0 dr D 4 R n0 r0 : .1=4/ m . 0 0
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 603 © Springer-Verlag Berlin Heidelberg 2015 604 F Solutions to problems
Solution to 1.4. Solution to 2.2. The total energy radiated throughout the (1) Differentiating the ansatz for r.t/, one finds rP D galaxy’s lifetime is E D Lt,wheret D 1010 yr is the ˛1 R D 2 ˛.r0=tf/.1 t=tf/ ,andr ˛.˛ 1/.r0=tf / assumed age. The energy generated is the mass that is ˛2 .1 t=tf/ . Inserting this into the equation of motion converted into helium in nuclear fusion, times the energy yields released per unit mass. The former is YM,whereY is the helium mass fraction generated by nuclear fusion. The 2 ˛2 D 2 2˛ 2 ˛.˛ 1/.r0=tf /.1 t=tf/ GM r0 .1 t=tf/ : energy released per unit mass is c ,where is the efficiency of this energy generating process, given by the ratio of the The powers of the time-dependent term must be the same binding energy per nucleon in helium ( 28 MeV=4 D one both sides, yielding ˛ 2 D2˛,or˛ D 2=3. 7 MeV) and the mass per nucleon, mnuc mp 938 MeV; Equating the prefactors then yields i.e., 0:008. Thus, the total energy released is E D s MY c2. Equating this to Lt, we obtain 3 2r0 GM 2r0 D ) tf D : Lt 1 Lˇ t 9t2 r2 9GM Y D D ; f 0 Mc2 3 Mˇ c2 !1 The solution r.t/has infinite radius for t ,butthe where we used the mass-to-light ratio. Inserting the values inflow velocity rP tends to zero as t !1.Att D 0, 33 33 for Mˇ 2 10 gandLˇ 3:8 10 erg=s, and using D P r.0/ r0, but the velocity r.0/ is finite and determined t 3:1 1017 s, we obtain Y 2:7 %. This value is lower D by M and r0.Fort tf, the radius shrinks to zero. by a factor of ten than the observed helium abundance. On N (2) Replacing the mass by the mean initial density of the the other hand, adding this value to the helium abundance sphere leads to s from BBN yields a result which is very close to the currently 1 observed helium abundance in local galaxies. t D ; f 6G N Solution to 2.3. According to the Kepler rotation, V.R/ D so that the time-scale t depends only on the initial p f G M.R/=R, a constant V implies M.R/ / R.The density. We can now compare t to the orbital time t D f orb relation between M and is 2R0=V0. Using (1.1), andp replacing M by the mean D N D Z density,p we get torb 3=.G /. Hence, tf=torb R 7 2 1=. 18/ 0:075. Thus, tf 1:75 10 yr. M.R/ D 4 drr .r/ : 0 (3) Using the mean matter density of the Universe, N D m, as given by (1.10), we obtain Differentiating this w.r.t. R yields dM=dR D 4 R2 .R/. / D 2 On the other hand, M.R/ R implies dM=dR const:,so D p 2 2 tf : that .r/ r D const:,or .r/ / r . 3 ˝mH0
Hence, t is very similar to the estimated age of the Solution to 2.4. Light at the Solar limb is deflected by f 00 Universe, H 1, and agrees with the age of the Einstein– ˛Oˇ D 1:74, which is far smaller than the angular radius 0 0 ˇ D 16 of the Sun. Hence, if we consider a cone of light de Sitter model (1.13), for which ˝m D 1. Indeed, the expansion history of the Einstein–de Sitter model rays with vertex at the Earth and an opening angle of ˇ, follows exactly the same equation of motion, except that this cone will continue to diverge after being deflected at the the relevant solution is an expanding one, r.t/ / t 2=3, Solar limb. If the Sun had a larger distance D, its angular 1 which is obtained from our infalling solution by inverting radius would be smaller, namely D .D=1 AU/ ˇ.If the arrow of time (note that the equation of motion is equals ˛Oˇ, the rays of the cone with opening angle at Earth invariant against t !t), and shifting the origin of the would be parallel after light deflection at the Solar limb, time axis. and if the distance was slightly larger they would converge after deflection and go through a common focus. If a source Solution to 2.1. The angular diameter is ı D 3476=385000 was placed at this focus, the Sun would then produce an 9:03 103. To convert this to degrees, we recall that Einstein ring. Displacing the source slightly, the ring breaks D 180ı,sothatı 9:03 103 .180ı=/ 0:517ı up into a pair of images, with angular separation 2. Thus, 310. The solid angle covered by the Moon is .0:517ı/2=4 the minimum distance for lensing is D D ˇ=˛Oˇ AU 0:21 deg2, so the fraction it covers of the full sky is (cf. 552 AU, and the image splitting at the minimum distance 00 problem 1.2) 0:21=41253 5:1 106. would be 2˛Oˇ 3:48. F Solutions to problems 605 p Solution to 2.5. Kepler rotation yields V.r/ D GM=r, 0:415:96 1 Lˇ I0 D Lˇ 10 175 : which we rewrite as .100 10 pc/2 pc2 s  à  à 00 1=2 1=2 GMˇ M r (2) At the distance of 16 Mpc, 50 correspond to hR V.r/ D c 4 kpc. The luminosity is obtained by integrating the c2 pc Mˇ 1 pc surface brightness over the disk,  à  à 1=2 1=2 Z M r 1 7 2:2 10 c : D R=hR D 2 Mˇ 1 pc L 2 dRRI0 e 2 hR I0 ; 0
6 With M 4 10 Mˇ, we obtain at r D 4 pc a rotational and by inserting numbers, velocity of 2:2 104c 66 km=s. Hence, at about this radius, the Keplerian rotation velocity around the black hole 2 175 Lˇ 10 L D 2.4 kpc/ 1:76 10 Lˇ : equals the velocity dispersion of the stellar cluster. pc2
Solution to 2.6. A light ray from the GC to us which is Solution to 3.2. According to the assumptions, the mass scattered in the screen at radius R has a geometric length function of the stars is n.m/ dm / m ˛ dm, with ˛ D 2:35. Ä Ä of L D L1 C L2,whereL1 is the length of the ray path from If half the mass is contained in stars with mass mL m us to the point R in the screen, and L2 the length of the path mm50,then from the GC to that point. Trigonometry then yields L D Z Z p p 2 m m 2 C 2 D C 2 C 2 2 U m50 D R D 1 .R=D/ D 1 R =.2D / , dmmn.m/D 2 dmmn.m/; where we made use of the fact that R=D 1 and used mL mL a first-order Taylor expansion of the square root. Similarly, D C 2 2 so that the result for mm50 is independent of the normalization L1 Dsc 1 R =.2Dsc/ . Thus, the total length of the ray path is of the mass function. Integration then yields  à 2˛ 2˛ D 2˛ 2˛ R2 1 1 mL mU 2 mL mm50 : L D L1 C L2 D D C Dsc C C : 2 D Dsc Solving for mm50 yields The light-travel time along this ray is L=c.Ifwedefinet as D 1=.˛2/ 2˛ C 2˛ 1=.2˛/ the excess of this light-travel time relative to a straight ray, mm50 2 mL mU ; we obtain  à  à or mm50 0:55 Mˇ. 2 2 2 R 1 1 R R0 R To calculate m , we need to satisfy t D C D L50 2c D Dsc 2c DDsc 2c D Z Z mU mL50 3 D 3 dmm n.m/ 2 dmm n.m/ : where in the final step we used D R0,orDsc R0. mL mL Hence, R2 2cDt. Differentiating both sides w.r.t. t,we get 2RRP D 2cD,orRP D cD=R. This apparent velocity is The same steps than lead to the result larger than the speed of light, since R D. If the scattering D 1=.˛4/ 4˛ C 4˛ 1=.4˛/ screen is located behind the Galactic center, the situation is mL50 2 mL mU ; very similar to the one described here. or mL50 46 Mˇ. We thus see that a stellar population Solution to 3.1. with such a mass spectrum contains most of its mass in the (1) We consider the central square arcsecond of the galaxy low-mass stars, whereas most of the luminosity is due to the as a source; its apparent magnitude (all magnitudes highest-mass stars. considered here are in the B-band) is m D 0 arcsec2 D 21:5. Thus, its absolute magnitude is Solution to 3.3. The volume contained in a solid angle ! M D m 5 log.D=10 pc/. The corresponding out to distance D is V D !D3=3.IfnP is the supernova event 0:4.M Mˇ/ luminosity of that source is L D Lˇ 10 D rate per unit volume, the observed SNe rate in the solid angle 0:4.mMˇ/ 2 P 3 Lˇ 10 .D=10 pc/ . The central square ! up to distance D is N DPnV D ! nDP =3. Inserting num- arcsecond corresponds to an area of .D 100/2,andso bers, NP D !105 Mpc3 yr1 .500 Mpc/3=3 D ! 417 yr1. 00 2 2 2 the central surface brightness is I0 D L=.D 1 / . Thus, Since 1 deg D =180, we can write ! D .!=deg /.=180/ , 606 F Solutions to problems so that NP 0:127 .!=deg2/ yr1. Thus, one needs to survey Hence, the magnification of all images is changed with 80 deg2 to find 10 nearby SNe per year. this new mass model; however, the magnification ratios between images stays the same, and thus the predicted Solution to 3.4. flux ratios. (1) The volume of a cone with height Dlim and opening solid angle ! is Solution to 4.1. At BBN, T 0:1 MeV 109 K 8 4 10 T0,whereT0 is the current temperature of the CMB. Z Dlim Hence, BBN happens at a scale factor of a 2:5 10 9. 2 D 3 BBN dr!r Dlim!=3 : The current baryon density is determined by (4.68) and the 0 2 critical density, b.0/ 0:02h cr, and the baryon density D D 3 (2) If the luminosity of a source is larger than L at BBN is then b.BBN/ aBBN b.0/. Using (4.15), we find 2 5 3 4 Smin Dlim, its flux is larger than Smin even out to b.BBN/ 2:5 10 g=cm . This density is many orders distances Dlim, and so they can be seen throughout the of magnitude lower than in the center of the Sun, or in other search volume. On the other hand, if their luminosity is stars. smaller than this value, theyp can only be seen out to a Nuclear burning in stars is slow because the thermal energy distance D D Dmax.L/ D L=.4Smin/. Thus, of protons (i.e., temperature) is too low to allow overcom- s ! ing the Coulomb barrier, that is the electrostatic repulsion between equally charged particles. Only very rarely does D L I Dmax.L/ min ;Dlim this process happen; quantum-mechanically, it occurs by 4Smin a process called ‘tunneling’. In contrast, the temperature ! during BBN was high, which makes it far easier to beat V .L/ D D3 .L/ : max 3 max electrostatic repulsion. Second, and most important, stars contain essentially no free neutrons, in contrast to the situ- (3) If ˚.L/dL is the number density of galaxies with ation at BBN; the formation of deuterium at BBN thus did luminosity within dL of L, then we observe not involve a Coulomb barrier, only the transformation of Z deuterium into helium. L 0 0 0 The energy density E released during BBN is given by the N.L/ D dL ˚.L /Vmax.L / 0 mass fraction of nucleons that ended up in helium, which is Y 0:25, times the binding energy per nucleon in helium, galaxies in the survey with luminosity Ä L. Differentia- corresponding to 7 MeV, times the number density of tion of this relation then immediately yields the desired nucleon nnucl. The latter is given by the baryon density at result. BBN, b.BBN/, divided by the mass of the nucleon, which is about 940 MeV=c2. Thus, Solution to 3.5. 7 .0/ (1) The deflection angle corresponding to the surface mass D MeV b 2 E Y 3 c : density (3.84) consists of two terms. The deflection 940 MeV aBBN due to the second term in (3.84) is simply ˛.Â/.The deflection due to the first term can be calculated from The energy density of photons at BBN is given by D 4 2 (3.70). Together, we find ˛.Â/ D .1 / C ˛.Â/. ˝CMB craBBNc . Hence, (2) The corresponding lens equation reads ˇ D ².Â/ D 2 ŒÂ ˛.Â/. Dividing the lens equation by , we obtain E 3 ˝bh 10 7:4 10 Y aBBN 3:7 10 ; ˇ D ˇ= D  ˛.Â/. That means that the new ˝CMB mass distribution yields the same image positions as the original mass model, provided the source position where we made use of the results from the first part of this is changed from ˇ to ˇ. Since the source position in problem and used (4.27). Hence, the energy released during unobservable, this shift in the source plane can not be BBN is totally negligible compared to the energy of the observed. photon gas. (3) The magnification is given by (3.68), and thus reads for the modified mass model Solution to 4.2. From the general definition (4.34), we need to calculate ˇ Â Ãˇ ˇ Â Ãˇ ˇ ˇ ˇ1 ˇ ˇ ˇ1 ˇ @ ˇ 1 ˇ @ ˇ R D ˇdet ˇ D ˇdet ˇ D : a 4G @ 2 @ 2 D C 3P =c2 : a 3 F Solutions to problems 607 P D We have (atP the current epoch) cr i ˝i ,and valid, the time-dependence on the r.h.s. must be of the form 2 D 2 P=c cr i ˝i wi , where the sum extends over all cosh .t=ta/; this can be achievedp by choosing v0 such that 2 D D N species. Hence,P with the definition of cr, we obtain .˝=˝m/v0 1,orv0 ˝m=˝. The prefactors R D 2 C a=a .H0P=2/ i ˝i .1 3wi /. Using (4.34), this yields on both sidesp also must agree, which then determines D C D q0 .1=2/ i ˝i .1 3wi /. With a pressureless matter ta 2=.3H0 ˝/. component (w D 0) and the vacuum energy with w D1, Hence, the final solution reads (4.35) is recovered.  à  p à ˝ 1=3 3H ˝ t a.t/ D v2=3 D m sinh2=3 0 : Solution to 4.3. Since aP D H.a/a D H0E.a/a,an ˝ 2 expansion can change into a contraction (or the opposite) D D 2 D 3 C only when H.a/ 0.For˝ 0,wehaveE ˝m=a Considering the case t ta and makingp use of sinh.x/ 2 3 2 2=3 .1 ˝m/=a D ˝m.1 a/=a C a . This expression is x for x 1,wefinda D 3H0 ˝m t=2 ,which 2 always positive for 0
2 Consider two light rays separated by a small angle at we can write K D 2GM=r0 Pr .0/. The solution D 2=3 the observer. For a flat universe, the comoving separation found in problem 1.4 wasr r0.1 t=tf/ ,which L.a/ D x D 2=3 2 2 D between these two rays is then given by . yields K .1 t=tf/ 2GM=r0 4r0 =.9tf / 0, The physical separation is then aL.a/. According to the when using the value for tf derived in problem 1.4. This definition of DA,wethenhaveDA D a L.a/= D x=.1Cz/, solution corresponds to one where the sphere at t D 0 which reproduces (4.54) for the case K D 0. has an initial infall speed, such that the total energy of the sphere is zero, in full analogy to a time-reversed Solution to 4.7. Einstein–de Sitter model. Setting the initial velocity to (1) Consider the case K>0first. We note from (4.85) that zero, r.0/P D 0, yields K D 2GM=r0 >0. In the context the parameter corresponding to t D t1 is 1 D 0, of the equation of motion discussed above, the free-fall since . sin / is a monotonically increasing function. time is the time between the maximum expansion and D D D 3=2 The first of (4.85) then yields f.t1/ 0, i.e., the initial the time of collapse,q tff tcoll tmax CK =2, D D condition f 0 at t t1 is satisfied by (4.85). Denoting D 2 3 which yields tff r0 =.8GM /. Replacing M by the derivatives w.r.t. t by a dot, those w.r.t. by a prime, mean density of the sphere, we arrive at (4.88). (4.84) reads fP2 D C=f K. Differentiation yields f 0 D C sin =.2K/, t 0 D C.1 cos /=.2K3=2/,sothat Solution to 4.8. fP D f 0=t0 D K1=2 sin =.1 cos /. The l.h.s. of (4.84) (1) Conservation of kinetic plus potential energy (per then becomes 2 unit mass) yields rP =2 GME=r D const:,or 2 2 2 C rP D 2GME=r K. At initial time t0, r.t0/ D rE, P2 sin 1 cos 1 cos f D K DK DK : r.tP / D v ,sothatK D v2 v2, which is assumed to .1 cos /2 .1 cos /2 1 cos 0 0 esc 0 be positive in the following. The r.h.s. of (4.84) is C=f K D 2K=.1 cos / K, (2) The equation of motion has the form (4.84), with D which is seen to agree with the above expression. Hence, C 2GME, hence the solution (4.85) applies. Without D (4.85) indeed solves (4.84) with the correct initial con- loss of generality, we set t1 0, and denote the time dition. The case K<0can be treated in the same way. when the object leaves the Earth surface by t0.The For K D 0, (4.87) yields f.t1/ D 0, as required, and corresponding parameter value 0 is found from the first P 1=3 1=3 D 2 f D .2C=3/ .t t1/ . Simple algebra then shows of (4.85), rE C.1 cos 0/=.2K/ C0 =.4K/, that fP2 C=f D 0. Considering the case K>0again, where we used the leading order of the Taylor f attains a maximum where cos has its minimum, expansion and assumed that 0 1, which needs to which occurs for D ; hence, fmax D C=K, at time be verified. Writing the initial velocity as a fraction 3=2 D tmax D t./ D t1 C C=.2K /. Furthermore, for of the escape velocity, v0 vesc, we see that 2 D D 2 2 D 2 D 2, f D 0, which happens at time tcoll D t.2/ D 0 4KrE=C 4.1 /vescrE=.2GME/ 4.1 /. 3=2 t1 C C=K . Hence, 0 1 if the initial velocity is sufficiently D D P2 D 2 C close to the escape velocity, 1 1,which (2) For ˝ 0 ˝r, (4.33) reads a H0 Œ˝m=a .1 ˝m/, which is seen to have the same form as will be assumed in the following (corresponding to D 2 D 2 the assumption that the flight is ‘long’). Then using (4.84), with C H0 ˝m and K H0 .˝m 1/. Setting t1 D 0 then yields a.0/ D 0, the correct the second of (4.85), we obtain in the same manner D 3=2 3 3=2 D initial condition for Friedmann expansion. Hence, with t0 C.0 sin 0/=.2K / C0 =.12K / 2.1 2 3=2 3=2 3=2 D 2 3=2 these parameter values, (4.85) describes the expansion / CK =3.SinceC=K .1 / rE=vesc, D for ˝m >1, (4.86) the expansion for ˝m <1,and we finally obtain t0 .2=3/rE=vesc. 2=3 D (4.87) yields the EdS case, a.t/ D .3H0t=2/ .For (3) The return time tret is given by tcoll 2t0, with tcoll 3=2 D 2 3=2 ˝m >1, the previous results then show that amax D C=K .1 / rE=vesc. Provided 1 C=K D ˝m=.˝m 1/, occurring at time tmax D 1, tcoll t0, and so we can approximate tret tcoll. 3=2 3=2 2 3=2 C=.2K / D =.2H0/˝m=.˝m 1/ , and collapse Inserting numbers, we obtain tret 1800 s=.1 / . D D happens at tcoll D 2tmax. Setting tret 1 d, we find 0:93, whereas for tret (3) Differentiation of (4.84) w.r.t. t yields 2fPfR D 1 yr, 0:994. Hence, if one wants to have a really C f=fP 2,orfR D.C=2/=f 2. The equation for long flight, the initial velocity must be extremely well the radius of the sphere is rR DGM=r 2,sothatwe tuned. can identify f with r,andC D 2GM . The constant K is proportional to the (negative of the) total energy Solution to 4.9. The momentum behaves like p D mv / of sphere, K=2 DPr2=2 GM=r,asthesumof .1 C z/,wherem is the rest mass of the (non-relativistic) specific kinetic and potential energy. With r0 D r.0/, particle, and v its velocity as measured by a comoving F Solutions to problems 609 p observer. The temperature Tb is related to the mean velocity Specializing the last result to z 1,using1= 1 C z D 2 / 2 / 3 dispersion as .3=2/kBTb mv =2. Thus, Tb v 1 z=2, we then obtain N .4=3/.c z=H0/ ncom, .1 C z/2. which is the number of objects in a sphere of radius c z=H0. Solution to 4.10. Consider a cosmic epoch, characterized by the scale factor a, at which the neutrinos were ultra- Solution to 4.12. The reason for the absence of H in D 0 relativistic; their momentum then was p E=c. The mean (4.61) is that in the Friedmann equation (4.18) curvature and energy of the thermal distribution is about 3T,whereT cosmological constant can be neglected at early times, and 0:7T , according to (4.62). With T D 2:73 K=a 2:5 2 4 2 thus .a=a/P / r / T .a/.Inotherwords,a=aP D CT , 4 4 10 eV=a,wethengetpa 5 10 eV=c. This product is where the constant C depends on the number of relativistic conserved, as shown by (4.47). When the temperature of the species only—see (4.60). The first law of thermodynamics universe drops below the neutrino rest mass, the momentum (4.17) then yields that T / 1=a [see (4.24)], which leads D is p mv. Thus, we obtain for the characteristic velocity to T=TP DPa=a DCT2, without any reference to the of cosmic neutrinos current universe – neither to its expansion rate, nor to its  à current temperature. The foregoing equation can be solved 2 1 m c ˛ P v .1 C z/ 150 km=s : with the ansatz T.t/ D xt , which yieldsp T=T D ˛=t D 1 eV Cx2t 2˛, and thus ˛ D1=2, x D 1= 2C . The helium abundance depends on the density of baryons (given that the density of photons as a function of temper- Solution to 4.11. 3 2 ature is known). But b D ˝b cr.1 C z/ / ˝bh .1 C D C 3=2 (1) From (4.56), we see that t t0.1 z/ . The look- z/3. Hence, the combination ˝ h2 determines the physical D D C 3=2 b back time is .z/ t0 t.z/ 1 .1 z/ t0. baryon density. The redshift where .z/ D t0=2 is then determined by C 3=2 D D 2=3 .1 z/ 2,orz 2 1 0:59. Solution to 4.13. The scattering optical depth is given by (2) The volume of the spherical shell is the product of the line-of-sight integral over the product of the Thompson the surface of the sphere at redshift z and the physical scattering cross section T and the number density ne of free thickness c dt of the shell, corresponding to the redshift electrons, interval dz. By definition of the angular diameter dis- Z 2 tance, the surface is 4DA.z/,sothat D T c dtne : ˇ ˇ ˇ ˇ 2 ˇc dt ˇ dV D 4D .z/ ˇ ˇ dz : D A dz From problem 4.11 we know that dt dz=.z H/ (in this problem we set 1 C z z, since all the contributions to Furthermore, from dt D da=.a H/ and da=a D the integral comes from z & 800). Since recombination dz=.1 C z/,wefindthatdt Ddz=Œ.1 C z/H.Using happens in the matter-dominated epoch, we have z H D 1=2 1=2 3=2 5=2 / 5=2 (4.57) and H D H0.1 C z/ , valid for the EdS model, H0˝m z h˝m z . The number density of free one finally finds electrons is equal to the number density of protons np and 3 the ionization fraction x.Fornp,wehavenp / ˝b crz /  à  à 2 3 c 3 1 1 2 ˝bh z ,sothat V D 16 1 p : H .1 C z/9=2 C Z 0 1 z z ˝ h2y3 .z/ / dy b x.y/ : R 1=2 5=2 D D 0 h˝m y (3) TheR number of objects is N dVcom ncom n dV , where the comoving volume element com com Using (4.72), we see that the ˝’s and h’s drop out, and we dV is related to the physical (proper) volume element com Z 3 obtain z by dVcom D .1 C z/ dV . Thus .z/ / dyy13:25 ;  à Z  à 0 c 3 z dz0 1 2 N D 16 n 1 p which yields the correct functional dependence of (4.73). com 0 3=2 0 H0 0 .1 C z / 1 C z  à  à Solution to 5.1. 32 c 3 1 3 D ncom 1 p : (1) The integral over the emissivity of a single electron 3 H0 1 C z f.=c/ over all frequencies is 610 F Solutions to problems Z Z  à  à 1 1 3=4 3=4 D / 2 hP hP r r df.=c/ c dxf.x/ ; x D D DW ; 0 0 kBT.r/ kBT0 r0 0 r0 where we used (5.3). This dependence on the Lorentz where in the last step we defined 0, we see that r=r0 D factor is thus the same as in (5.5). 4=3 4=3 5=3 8=3 x .=0/ . Therefore, r dr / x .=0/ dx,and (2) The distribution of relativistic electrons is N./d D s 2  à a E D m c a / 1=3 Z 1 d ,since e ,and is a constant x5=3 A. The synchrotron emissivity of this distribution is L / dx : ex 1 obtained by integrating the emissivity of a single electron 0 0 over the electron distribution,  à Z 1 Z 1 Solution to 5.4. At an accretion rate of mP , the growth rate s P D dN./ f .=c/ D a d f ; of the black hole is M D .1 /mP , since the fraction of 2 0 0 0 the accretion rate is converted into luminosity, and thus does 2 not contribute to the increase of the black hole mass. With where we defined 0 D c= D 3eB=.4mec/. 2 (5.15), one then finds Changing thep integration variable to x D =.0 /, with 3=2 d D .1=2/ =0 x dx then yields P 1 L 1 L M M D L D ;  à 2 edd .s1/=2 Z 1 c Ledd Ledd tgr a .s3/=2 D dxx f.x/: 2 0 0 and the relation (5.47) is the solution of this equation. Inserting the specific values L D Ledd, D 0:1,andt D Since the final integral is frequency-independent, we find 9 D 18 7 D 10 yr, we find M.t/ M.0/ e 6:6 10 M.0/,or that the emitted spectrum is a power law with index ˛ 8 M.t/ 6:6 10 Mˇ. .s 1/=2. Solution to 5.5. Solution to 5.2. (1) We assume all clouds to be at the characteristic distance (1) Replace B2 D 8U in (5.5) and use the definition B r from the SMBH; each cloud covers a solid angle of (5.23) for to arrive at the expression given. T r2=r2. The covering fraction is obtained by summing (2) Consider first the case that all photons have the same c the solid angle over all Nc clouds and dividing by 4,to energy E . The energy loss of a relativistic electron scat- D 2 D 2 find fcov .Nc=4/ .rc=r/ . tering a photon then is on average E .4=3/ E , 1 (2) The volume filling factor fV is the ratio of the total and the time between scatterings is t D n c T . 3 volume of the clouds, Nc.4=3/rc to that of the BLR, With U D n E and dE=dt D E= t, the desired 3 3 .4=3/r , fV D Nc.rc=r/ . expression is obtained. Since this expression no longer 6 2 1=3 (3) Using fV D 10 , one obtains .rc=r/ D 10 Nc . refers to the photon energy (or frequency), but only to 2 From fcov D 0:1 we get Nc D 0:4.rc=r/ . Combining the energy density of photons, the same result is obtained 10 these two expressions, we obtain Nc D 6:4 10 for a spectral distribution of photons. 6 10 and rc D 2:5 10 r D 2:5 10 cm. The total gas mass of the clouds is the product of the clouds’ Solution to 5.3. We write the temperature profile (5.13) as volume times the electron density (that then yields the D 3=4 T.r/ T0.r=r0/ . The specific intensity is that of a total number of electrons in the clouds) times the average blackbody with temperature T.r/,sothat mass per electron, which is about the proton mass (since Ä Â Ã there are about as many electrons as nucleons). Hence, 1 h 3 3 P Mc D ne Vc mp. The volume is Vc D Nc.4=3/r I.r/ / exp 1 : c 42 3 52 24 kBT.r/ 4 10 cm ,sothatMc D 4 10 1:67 10 g 28 5 6:7 10 g 3:4 10 Mˇ. Hence, the total gas mass The emitted luminosity L is then the integral of the specific in the BLR is very small indeed. intensity over the surface of the disk,
Z 1 Solution to 5.6. (1) According to the assumption, the optical light from the L / drrI.r/ ; 0 AGN is where we neglected boundary effects by setting the integra- 37 L M LAGN;opt D 0:1L D 1:3 10 erg=s tion limits to 0 and 1. Writing the exponent as Ledd Mˇ F Solutions to problems 611
L M Solution to 6.2. The observed bolometric flux is, according 3400Lˇ ; D 2 Ledd Mˇ to the definition (4.50), S L=.4DL/, and the angular radius is D R=DA. Hence, the surface brightness behaves D 2 2 2 2 where we used the Solar luminosity Lˇ 3:85 like I D S=. / / .L=R /.DA=DL/ D .L=R /.1 C 38 4 10 erg=s. The optical light from the host galaxy is z/ . For the specific surface brightness I, the redshift- 1 L D M .M=L/ D .M=Mˇ/Lˇ.M=L/ˇ=.M=L/. dependence can be different, depending on the spectral prop- 3 Using M D 10 fsphM, we then find erties of the source, according to the necessary K-correction 2 (see Sect. 5.6.1). With I D S=. / and S from (5.42), L ; L .M=L/ D 2 C 1 ˛ / AGN opt we obtain S L=.4DL/.1 z/ , yielding I 3:4 fsph : .3C˛/ L Ledd .M=L/ˇ .1 C z/ .
(2) With an Eddington ratio of 0:1 and a typical mass- Solution to 7.1. If is assumed to be spatially constant, the to-light ratio of 3 in Solar units, the light from the continuity equation becomes @ =@t C r v D 0.Since AGN is comparable to that of the host galaxy if the the first term is independent of r, so must be the second, spheroidal fraction is close to unity. For late-type galax- which immediately implies that r v can depend only on ies, where fsph is considerably smaller, the host galaxy time. Inserting v.r;t/ D H.t/r then yields P C 3 H D 0, will typically dominate the optical emission—this is one or = P D3a=aP . By insertion, we see that the solution with of the reasons why a complete census of AGNs in the .t0/ D 0 is given by (4.11). optical is difficult to achieve. The AGN can outshine the Since r2jrj2 D 6, a solution of the Poisson equation is host galaxy only for apparently large Eddington ratios  à (e.g., when beaming is involved), or if the SMBH is 2 ˚.r;t/D G .t/ r2 ; considerably larger than the assumed scaling. 3 6
r D r Solution to 5.7. If the tidal disruption distance Rt D so that ˚ Œ.4=3/G =3 . For the terms on the 1=3 R.M=M/ is smaller than the Schwarzschild radius of l.h.s. of the Euler equation, we find the SMBH, the star will be swallowed by the black hole  à @v aR aP2 before it can be disrupted. Hence we require Rt >rS. D P r D r 3=2 1=2 2 3=2 H ; This yields M
is a solution of (7.15). We first note that Inserting the values for the Schwarzschild radius of the Sun and c=H0,wefind da dI 1 1 IP D D aH D ;  à 3 3 2 2 1=3 dt da a H a H M 3 1=3 2=3 r200 ˝m.1 C z/ C ˝ h Mˇ so that  Ã1=3  à  à 3 105 81 1054 cm : P P H P 1 DC D C HI C D C HI C : 200 a2H 2 a2H With the estimates 3 81=200 1:2, the number in Furthermore, the last parenthesis yields approximately 1:2 1059 D ! 120 1057.Since53 D 125, the third root of 120 is very P P R R H 2aP H close to 5, so that DC D C HI C a2H 2 a3H a2H 2  à 1=3 19 M 3 1=3 2=3  à r200 510 cm ˝m.1Cz/ C˝ h 2 Mˇ D C HIR : 2  à a 1=3 M 3 1=3 2=3 16pc ˝m.1Cz/ C˝ h : Collecting terms, Mˇ (E.2) R P DC C 2HDC 4G DN C D D 1 Thus, at z 0, we obtain r200 160h kpc and R P 1 D CI H C 2HH 4G HN D 0; r200 D 1:6h Mpc for the galaxy and cluster mass halo, respectively. where the final equality was shown in the first part of this At z D 2, we see that the matter term in the bracket 3 dominates, since ˝m.1 C z/ D 0:3 27 D 8:1 ˝, problem. P 3 1=3 3 (3) For the EdS model, we have H D 2=.3t/, H D so that ˝m.1 C z/ C ˝ 1=2,since2 D 2=.3t2/ and HR D 4=.3t3/. Inserting these expression 8. Thus, the virial radius at z D 2 is about half the as D D H into (7.15), using H0 D 2=.3t0/,showsthat size of that today, for fixed mass. Hence, halos of a it is satisfied, i.e., H is a solution of the growth equation. given mass are smaller at higher redshift, as expected, Furthermore, specializing (7.17) to the EdS parameters since the definition of a halo—‘mean density equals 200 yields times critical density’—together with the increase of the critical density with redshift implies that higher-redshift  à Z  à  à t a t t 2=3 halos have a larger mean density, i.e., they are more / 0 0 03=2 / 0 5=2 / DC da a a : compact. t 0 t t0 For V200, we start from (7.58) and use the preceding 2=3 result: Inserting D D .t=t0/ into (7.15) shows that it indeed solves the growth equation. V200 D 10 H.z/r200 p 1 1 3 Solution to 7.3. D 10 h 100 km s Mpc ˝m.1Cz/ C˝  à (1) The relation between virial mass and virial radius is 1=3 M 3 1=3 2=3 given by (7.56), so that 16pc ˝m.1Cz/ C˝ h (E.3) Mˇ  à  à 1=3 1=3 GM 3 km M 3 1=6 1=3 r200 D : D 1610 ˝m.1Cz/ C˝ h : 100 H 2.z/ s Mˇ
Using the Hubble function for a flat universe, and trans- Hence, our two halos have virial velocities of 160 km=s forming the product GM into the Schwarzschild radius and 1600 km=s, respectively, at redshift zero, indepen-p yields dent of h.Atz D 2,the˝-dependent term is 2  à (recall the earlier estimate), hence the corresponding 2 1=3 2GMˇ M c virial velocities are 225 km=sand2250pkm=s, respec- r D : 200 2 2 C 3 C tively. That they are higher by a factor 2 is already c Mˇ 200 H0 Œ˝m.1 z/ ˝ F Solutions to problems 613
clear from Kepler’s law, since the radius is smaller by 3=2 5=2 the matter term dominates, i.e., I .2=5/˝m a . a factor 2, for fixed mass. Together, DC / a for sufficiently small a. (2) The mass M contained in a proper volume Vprop at (2) To obtain the next-order term, we write for a flat universe DN C redshift z in a homogeneous Universe is M m0.1 Z 3 a 0 z/ Vprop. Correspondingly, the mass in the comoving vol- da I.a/ WD ume V D .1 C z/3V is M DN V , independent 3=2 com prop m0 com 0 ˝ =a0 C ˝ a02 D 3 m of redshift. Writing Vcom .4=3/R and using the Z definition of the virial radius, specialized to the current 1 a da0 a03=2 D epoch, we obtain 3=2 03 3=2 ˝m 0 1 C .˝=˝m/a Z  à 4 4 1 a 3 ˝ R3 ˝ D r3 200 ; 0 03=2 03 m cr 200 cr 3=2 da a 1 a 3 3 ˝ 0 2 ˝m m  à 1 2 15 ˝ where r200 is the virial radius of the halo of mass M D 5=2 3 3=2 a 1 a ; today. Thus, ˝m 5 22 ˝m  à 1=3 where we performed a Taylor expansion of the integrand D 200 R r200 8:7r200 : in the second step. A similar Taylor expansion of the ˝m Hubble function yields Thus, the mass of our two halos were assembled from a p  à volume corresponding to a sphere of comoving radius of H.a/ ˝m 1 ˝ 1 C a3 : 1 1 3=2 1:4h Mpc and 14h Mpc, respectively. H0 a 2 ˝m (3) According to (7.38), the scale factor a at which a pertur- bation of length scale L enters the horizon is given as Multiplying these two results then yields Z  à a 0 c da 2 ˝ 3 6 L D p p DC / a 1 a C O.a / : 0 11 ˝ H0 ˝m 0 a C aeq m 2c p p If we consider a significant deviation from the linear D p a C aeq aeq H0 ˝m behavior to occur when the second term in the paren- r Âr à thesis becomes of order 0.1, then we request a3 . 2c aeq a D 1 C 1 (E.4) 0:1.11=2/.˝m=˝/. Taken the parameters which apply H0 ˝m aeq to our Universe, ˝m=˝ 3=7, this becomes a . 0:6. c a c a This estimate is in concordance with the behavior of the p D p ; H0 ˝maeq H0 ˝r dashed curve in Fig. 7.3.
where we made a first-order Taylor expansion, assuming Solution to 8.1. a aeq, and used the definition of aeq. Note that we (1) The mean number density of observed galaxies is, have just rederived (4.76). Using (4.28), this becomes according to (8.50), Z a D 1 p 5 L 3000h Mpc 4:6 10 a Mpc : nN D dx3 .x3/ nN3.x3/; 42 106h2
Thus, the perturbation that eventually led to the for- so that the probability px.x3/ dx3 for an observed galaxy mation of our galaxy-mass halo entered the horizon at to have distance within dx3 of x3 is proportional to 6 1 a 3 10 h , that corresponding to our cluster-mass .x3/ nN3.x3/ dx3. Normalizing px.x3/ to unity yields halo at ten times this scale factor. Note that both entered (8.51). the horizon in the radiation-dominated epoch, a aeq, (2) Substituting .x3/ by px.x3/ in (8.50) yields so that indeed (4.76) applies. Z n3.fk .x3/Â;x3/ n.Â/ DNn dx3 px.x3/ Solution to 7.4. nN3.x3/ Â Z Ã (1) The growth factor (7.17) forpa 1 is obtained from 3=2 DN C Â the product of H.a/ H0 ˝ma and the integral n 1 dx3 px.x3/ıg.fk.x3/ ;x3/ ; I in (7.17), in which for a 1 (and thus a0 1) 614 F Solutions to problems
1=2 whereweusedthatıg.x/ D Œn3.x/ Nn3.x3/=nN3.x3/. / / / or ne DA . The estimated gas mass is Mgas neV (3) We first define the number density contrast 5=2 DA . The total mass within as determined from (6.37) can Z be written as n.Â/ Nn ın.Â/ D D dx3 px.x3/ıg.fk.x3/Â;x3/: nN dln.n T/ M / TR e : dlnR In analogy to (7.27), the angular correlation function w
is defined as Varying DA adds a constant to ln R and thus does not change the derivative; hence, M / DA. Together with the previous hı ./ı . C Â/i D w.jÂj/: / 3=2 n n result, we find fgas DA .
We next define the redshift probability distribution p.z/ Solution to 8.3. We start with the Friedmann equation of the galaxies, given in terms of px.x3/ by p.z/ dz D written as (4.14), px.x3/ dx3, where the function x3.z/ is given in (4.53); however, we do not need to use this relation explicitly. aP2 D .8G=3/ .a/ a2 Kc2 ; Then, Z Z ˝ ˛ and subtract from it the same equation specialized to the D hın./ın. C Â/i D dz1 p.z1/ dz2 p.z2/ ıg ıg current epoch (a 1) to obtain Z Z P2 2 D 2 a .8G=3/ .a/ a H0 .8G=3/ 0 ; D dz1 p.z1/ dz2 p.z2/g ;
so that the constant K is eliminated. With cr.a/ D 2 where we made use of the definition (7.27) of the 3H .a/=.8G/ and ˝0.a/ D .a/= cr.a/, this can be correlation function g, and where the argument of g rewritten as depends on the separation of the two points characterized by the directions and C  and redshifts z . For this, 2 2 D 2 D i a H .a/ Œ1 ˝0.a/ H0 Œ1 ˝0.a 1/ ; we can either use the comoving separation (as was done during most of this and the previous chapter), or use the which reproduces (4.81), with F given by (4.82), or proper separation—they just differ by a factor .1 C z/.  à 1 The correlation g is assumed to be zero unless the two ˝ ˝ ˝ F.a/ D r C m C Œ1 ˝ .a D 1/ C DE ; galaxies are close in space. In particular, for a non- a2 a 0 a.1C3w/ zero correlation the two galaxies need to have a similar redshift. We thus define z1 D z C z=2, z2 D z z=2, where we used (8.49) but included the curvature term. Since where z D .z1 C z2/=2 is the mean redshift and we w < 1=3, the final term cannot increase as a ! 0. Hence, assume that g vanishes unless j zjDjz2 z1j as in the case of a cosmological constant, for early times the z. Then approximating p.z1/ p.z/ p.z2/,and radiation term dominates and the argument from Sect. 4.5.2 replacing the integration over z1 and z2 by one over z remains unchanged. and z, we obtain (8.17), where the separation in the argument of g is valid provided the angular separation Solution to 8.4. Writing .r/ D .r=r0/ , Limber’s equa- 1, so that we can use tan . tion (8.17) yields
2 Z Z 1 Â 0 Ã=2 Solution to 8.2. We have LX D 4D SX D 4.1 C 2 2C 2 2 L 2 DA.z/ D . z/ 4 2 D z / D S , where we used (4.52). Since X-ray emission is w./ dzp .z/ d. z/ 2 ; cl A X 1 r / 2 / 3 0 a two-body process, we have LX ne V ,whereV R D 0 is the volume of the region considered, and R DA.The where D D dD=dz. Substituting z D yand replacing temperature of the gas can be determined independent of the the inner integral by one over y with d. z/ D dy imme- distance, and just merely enters in the constant of proportion- diately yields the scaling w./ / .1/. With the same ality. Combining the two expression for L ,wefind X method applied to (8.13) it is readily shown that wp.rp/ / r . 1/. 2 / / 2 3 p DA LX neDA ; Index
Abell catalog. See Clusters of galaxies jets (see jets) Abell radius, 280 LINERs, 222, 244 AB magnitudes, 586 luminosity function, 219, 263–268 4000 Å-break, 134, 142, 466–468, 485, 521 narrow line region (NLR), 243 absorption coefficient, 584 obscuring ‘torus’, 254, 261 absorption lines in quasar spectra, 268–271, 423, 465, 471 OVV (optically violently variable), 222–223, 255 classification, 269–270 polarization, 222 Lyman-˛ forest (see Lyman-˛ forest) QSO (quasi-stellar object), 221 metal systems, 269, 270, 465 composite spectra, 212, 213 accelerated expansion of the Universe, 185, 416 – radio-loud & radio-quiet dichotomy, 221 accretion, 11, 225–227, 300, 483 quasars, 10, 214–219, 221 Bondi–Hoyle–Lyttleton accretion, 226–227 radio emission, 215–218, 252 cold vs. hot accretion onto dark matter halos, 528 radio galaxies, 4, 222 efficiency, 226 broad-line radio galaxies (BLRG), 222 radiatively inefficient accretion, 226 narrow-line radio galaxies (NLRG), 222 spherical accretion, 226 radio lobes, 215 tidal disruption event (TDE), 262–263 relativistic iron line, 230, 246 accretion disk, 219, 225–226, 230, 233 Seyfert galaxies, 11, 212, 222, 230 advection-dominated accretion flow, 226 soft X-ray excess, 234, 245, 246 corona, 245 spectra, 219, 220 geometrically thin, optically thick accretion disk, 225–226 TeV radiation, 260 temperature profile, 225 Type 1 AGNs, 222 viscosity, 226 Type 2 QSO, 254, 509 Acceleration of particles, 217, 257, 311 unified models, 219–220, 252–262 acoustic peaks, 432 variability, 215, 222–224, 239, 245, 256 active galactic nuclei, 105, 211–271 wide angle tail sources, 304 broad-band energy distribution, 215 X-ray emission, 244–247, 252, 257, 508 active galaxies, 10, 105, 211–262 X-ray reflection, 246 absorption lines (see absorption lines in quasar spectra) X-ray selection, 265 accretion X-shaped radio sources, 550 quasar mode, 261–262, 540 active optics, 28 radio mode, 261–262, 540, 565 adaptive mesh refinement (AMR), 553 anisotropic emission, 252 adaptive optics, 34, 92 big blue bump (BBB), 234, 245 Advanced Camera for Surveys (ACS), 30, 471 binary AGNs, 248, 549 age-metallicity relation, 58, 65 black hole, 224–233 Age of the Universe, 4, 17, 42, 185, 284, 336, 453, 455 black hole mass, 248–252 air shower, 39 scaling relation, 250 AKARI, 27 black hole spin, 230 ALMA (Atacama Large Millimeter/sub-millimeter Array), 23, 487, BL Lac objects, 222–223, 255 494, 573 blazars, 223, 234, 255 alpha elements, 57, 65 broad absorption lines (BAL), 269, 270 Andromeda galaxy (M31), 15, 101 broad emission lines, 212, 214, 218, 238 Anglo-Australian Telescope (AAT), 28, 394 in polarized light, 252 angular correlations of galaxies. See Correlation function broad line region (BLR), 238–243, 252 Angular-diameter distance, 189, 190 classification, 219–223, 252 angular momentum barrier, 536 in clusters of galaxies, 303, 304 Angular resolution, 19 Compton thick AGN, 266 anisotropy parameter ˇ of stellar orbits, 114 Eddington ratio, 250, 262 anthropic principle, 207 energy generation, 144 anti-particles, 193 host galaxy, 219, 221, 223, 247–248, 252 APEX (Atacama Pathfinder Experiment), 22, 486, 491 ionization cone, 243 Arecibo telescope, 20
P. Schneider, Extragalactic Astronomy and Cosmology, DOI 10.1007/978-3-642-54083-7, 615 © Springer-Verlag Berlin Heidelberg 2015 616 Index
ASCA (Advanced Satellite for Cosmology and Astrophysics), 36 bulge, 64, 110, 116, 565 astronomical unit, 47 of the Milky Way, 64–65 asymmetric drift, 72 Bullet cluster, 305, 327 asymptotic giant branch (AGB), 593 nature of dark matter, 324 Atacama Cosmology Telescope (ACT), 441 Butcher–Oemler effect, 335, 336, 552 attenuation of rays, 506 Australian Square Kilometre Array Pathfinder (ASKAP), 575 Canada-France-Hawaii Telescope (CFHT), 28, 394 Canada-France Redshift Survey (CFRS), 394 Baade’s Window, 64, 89 CANDLES survey, 501 background radiation, 504–509 cannibalism in galaxies, 552 infrared background (CIB), 505–508 CCD (charge-coupled device), 2 of ionizing photons, 425 Center for Astrophysics (CfA)-Survey, 343, 392, 393 limits from -ray blazars, 506 Cepheids, 53 microwave background (see Cosmic microwave background) as distance indicators, 54, 76, 151 X-ray background (CXB), 508, 509 period-luminosity relation, 151 bar, 64, 103 Cerro Chajnantor Atacama Telescope (CCAT), 576 baryogenesis, 580 Chandra Deep Fields, 473, 474, 491 baryon asymmetry, 580 Chandra satellite, 36, 126, 246, 257, 299 baryonic acoustic oscillations (BAOs), 354–357, 395, 397–399, 430, Chandrasekhar mass, 58, 417 449 characteristic luminosity L, 155, 158 as ‘standard rod’, 397 chemical evolution, 58, 65, 142–144 in Lyman-˛ forest, 428, 429 Cherenkov radiation, 39 measurements, 399 Cherenkov telescopes, 39, 260, 508 baryons, 5, 194, 196, 198, 427, 454 CLASH (Cluster Lensing And Supernova survey with Hubble) survey, Baryon-to-photon ratio, 354 477 beaming, 255–256 clustering length, 491 Beppo-SAX, 38, 517 clusters of galaxies, 12, 273–336, 408–414 biasing, 394, 402–403, 413, 462, 570 Abell catalog, 279–281 of dark matter halos, 374, 403, 409, 462 Abell radius, 280 as a function of galaxy baryon content, 413, 458 color, 403 beta model, 298–300, 317 luminosity, 397, 402 brightest cluster galaxy (BCG), 285, 301, 330 Big Bang, 4, 16, 184, 185, 188 Bullet cluster, 305, 324, 327 blackbody radiation, 584–586 Butcher–Oemler effect, 335 energy density, 585 catalogs, 279–281, 309–310, 408 Black holes, 144, 229 classification, 282 in AGNs, 92, 224–233, 529 color-magnitude diagram, 283 binary systems and merging, 547–550 Coma cluster, 13, 279, 293 (see Coma cluster of galaxies) demography, 268 cool-core clusters, 302, 336 evolution in mass, 556–559, 565 cooling flows, 300–306, 322 formation and evolution, 540–541 cooling time, 300 in the Galactic center, 7, 92–99, 144, 232, 251, 267 mass cooling rate, 300 in galaxies, 2, 9, 144–148, 229, 247, 267, 484, 547–550 core radius, 287, 288 at high redshift, 534 as cosmological probes, 273, 408–414 kinematic evidence, 145 dark matter, 273, 290, 300, 413 mass growth rate, 233, 541 distance class, 281 mass in AGNs, 248–252 evolution effects, 335–336 radius of influence, 144 extremely massive clusters, 410 recoil, 548 feedback, 303, 304, 540 scaling with galaxy properties, 3, 146–148, 251, 252, 549, 556, 565 galaxy distribution, 286–288 Schwarzschild radius, 144, 224, 230 galaxy luminosity function, 329–330 BL Lac objects. See Active galaxies galaxy population, 552 blue-cloud galaxies, 140 gas-mass fraction, 413, 414 bolometric magnitude. See Magnitude HIFLUGCS catalog, 312–314, 414 Bonner Durchmusterung, 40 intergalactic stars, 291–293 BOOMERANG, 436, 437, 441 intracluster medium, 13, 273, 291, 293–311, 552 bosons, 192 large-scale structure, 413–414 bottom-up structure formation, 361 luminosity function, 310, 335 boxiness in elliptical galaxies, 114 mass calibration by weak lensing, 327, 380 BPT (Baldwin–Phillips–Terlevich) diagram, 243–244 mass determination, 13, 289–290, 296, 300, 313, 319, 323, 408 bremsstrahlung, 294–295 mass function, 411 brightest cluster galaxy (BCG), 285, 301, 330, 381 mass-luminosity relation, 313, 316 brightness of the night sky, 292, 465 mass-temperature relation, 312 broad absorption line QSO (BAL QSOs). See Active galaxies mass-to-light ratio, 290, 324, 412 brown dwarfs, 591 mass-velocity dispersion relation, 313 Index 617
maxBCG catalog, 284–286, 410 from pair counts, 349 mergers, 296 projected, 401–405 metallicity of ICM, 299, 552 related to biasing, 396, 462 morphology, 282 slope, 404, 458 near-infrared luminosity, 316 correlation length, 349, 402–404 normalization of the power spectrum, 359, 410, 414 of Lyman-break galaxies, 462 number density, 408–411, 422, 450 Silk damping, 430 numerical simulations, 409 Cosmic Lens All-Sky Survey (CLASS), 162 projection effects, 280, 309 cosmic luminosity density, 498, 499 radio relics, 310 cosmic microwave background, 4, 16, 174, 306, 429, 504 RCS surveys, 284 acoustic peaks, 432 richness class, 281 baryonic acoustic oscillations, 430 scaling relations, 311–317, 408 cosmic variance, 438 redshift dependence, 315 dipole, 16, 150, 399, 407 self-similar behavior, 316 discovery, 203, 429 selection effects, 409 fluctuations, 342, 346, 391, 429–444 sound-crossing time, 297 dependence on cosmological parameters, 433 statistical mass calibration, 408 discovery, 429 temperature profile, 299 foreground emission, 435, 442 Virgo cluster, 13, 279 gravitational lensing, 430, 434, 444 weak lensing mass profile, 376, 409 integrated Sachs–Wolfe effect (ISW), 430 X-ray radiation, 13, 293–306, 312–314 measuring the anisotropy, 16, 434–444 X-ray spectrum, 301 origin, 203 Y-parameter, 314–315 polarization, 388, 434–445 Zwicky catalog, 280 primary anisotropies, 429–430 COBE (Cosmic Background Explorer), 16, 24, 203, 429, 435, 438 redshift evolution, 187 cold dark matter (CDM), 351 Sachs–Wolfe effect, 429 substructure, 563 secondary anisotropies, 429–431, 433, 434 collisionless gas, 113 Silk damping, 430, 432 color-color diagram, 51–52 spectrum, 187, 509, 529 color excess, 50 Sunyaev–Zeldovich effect, 430, 434 color filter, 586–587 temperature, 187 color index, 50, 586 Thomson scattering after reionization, 430, 433 color-magnitude diagram, 49, 485, 589 Cosmic Origins Spectrograph (COS), 30 color temperature, 589 cosmic rays, 61–64, 217 Coma cluster of galaxies, 13, 279, 294, 298 acceleration, 63 distance, 155 energy density, 64 Sunyaev–Zeldovich observation, 308 GZK cut-off, 63 comoving coordinates, 177, 178, 344 ultra-high energy cosmic rays (UHECRs), 63–64 comoving observers, 178, 186 cosmic shear. See Gravitational lensing completeness and purity of samples, 280, 286 cosmic variance, 438–439 Compton Gamma Ray Observatory (CGRO), 38, 516 cosmic web, 365 Compton scattering, 232 cosmological constant, 5, 15, 179, 180, 415, 416, 437, 440, 455 inverse, 203, 232, 246, 258, 260, 306, 509 smallness, 456 Compton-y parameter, 307 the ‘why now’ problem, 456 concentration index of the NFW profile, 367, 376 cosmological parameters confusion limit, 505 consistencies and discrepancies, 448 continuity equation, 344 degeneracies, 445, 447 in comoving coordinates, 344 determination, 359, 391–455 convection, 591 standard CDM model, 446, 447 convergence point, 48 cosmological principle, 176, 177 cool-core clusters, 303 cosmology, 15, 18, 173–209, 341–388, 391–455 cooling diagram, 527 classification of cosmological models, 184–185 cooling fronts, 305 components of the Universe, 180–182 cooling function, 526 curvature scalar, 183 cooling of gas, 60, 524, 526, 530, 563 dark ages, 477 the role of molecular hydrogen, 530 density fluctuations, 177, 341–388, 427 and star formation, 530, 563 epoch of matter–radiation equality, 182 cooling time, 300, 527 expansion equation, 178, 180–186 Copernicus satellite, 35 expansion rate, 178, 186 correlation function, 348–350, 353, 400, 401, 409, 431, 462 homogeneous world models, 173–209 angular correlation, 405 Newtonian cosmology, 177–179 anisotropy, 401 radiation density of the Universe, 182 definition, 348 structure formation, 16, 17, 342, 523 homogeneity and isotropy, 348 tensor fluctuations, 451 of galaxies, 348, 396, 401, 568, 569 COSMOS survey, 41, 381, 422, 473, 487 618 Index
3CR radio catalog, 213 extinction and reddening, 50, 418, 486 curvature of the Universe, 441 gray dust, 418 infrared emission, 60, 105, 125, 234, 254, 481, 486 warm dust, 234, 254, 482, 486 dark energy, 5, 180, 415, 416, 454–458, 580 dust-to-gas ratio, 51 equation-of-state, 456 dwarf galaxies. See galaxies Dark Energy Survey (DES), 577 dynamical friction, 290, 291, 542, 548 dark matter, 3, 77, 124, 198–201, 455, 579–580 dynamical heating, 552 in clusters of galaxies, 13, 273, 300 dynamical instability of N -body systems, 98 cold and hot dark matter, 351–352, 523 dynamical pressure, 55 filaments, 327, 328 in galaxies, 9, 122, 124 seen in ‘bullet clusters’, 324 early-type galaxies, 103 in the Universe, 17, 198, 346, 429 ecliptic, 47 warm dark matter, 385 Eddington accretion rate, 232, 233 dark matter halos, 77, 124, 358–361, 365–387, 523 Eddington luminosity, 230–233, 250, 252, 483, 516 angular momentum, 372 Eddington ratio, 250, 268 biasing, 374–375 effective radius Re, 65, 108, 130 contraction of gas, 536 effective temperature, 589, 590 cooling of gas, 526–528 Effelsberg radio telescope, 20 Einasto profile, 370 Einasto profile, 370 gas infall, 525–526 Einstein–de Sitter model. See Universe mass function, 523 Einstein observatory, 36, 310 Navarro–Frenk–White profile, 367–370 Einstein radius E. See gravitational lensing number density, 359–361, 367, 408–411, 529 elementary particle physics, 19, 193–194 shapes, 372 beyond the Standard Model, 199 spin parameter, 373, 536 elliptical galaxies. See galaxies stellar mass-halo mass relation, 560, 568 cored profile, 109 substructure, 381–387 emission coefficient, 584 universal mass profile, 367–372, 376 energy density of a radiation field, 583 virial radius, 370, 377 energy efficiency of nuclear fusion, 224 virial temperature, 525 epoch of matter–radiation equality, 182 weak gravitational lensing, 375–381 equation of radiative transfer, 49, 584 deceleration parameter q0, 185, 209, 422 equatorial coordinates, 45 declination, 46 equivalent width, 218, 424 DEEP2 redshift survey, 394 eROSITA space mission, 457, 579 deflection of light. See gravitational lensing escape fraction of ionizing photons, 532, 535 density contrast, 342, 357 Euclid space mission, 457, 577, 578 density fluctuations in the Universe, 341–388 Euler equation, 344 origin, 387–388, 451 in comoving coordinates, 345 density parameter, 15, 179, 182, 185, 206, 412, 422, 429, 437, 454 European Extremely Large Telescope (E-ELT), 577 as a function of redshift, 206 expansion rate. See cosmology deuterium, 197 Extended Medium Sensitivity Survey (EMSS), 310 primordial, 198 extinction, 49, 418 in QSO absorption lines, 198 coefficient, 50, 588 de Vaucouleurs law, 65, 108, 117 and reddening, 49, 418 diffraction limit, 19 extremely red object (ERO), 484–486, 521 Digitized Sky Survey (DSS), 40 diskiness in elliptical galaxies, 114 distance determination, 148, 309 Faber–Jackson relation, 130 of extragalactic objects, 128, 148–155 Fanaroff–Riley classification, 215, 222 within the Milky Way, 46–54 Faraday rotation, 62, 256 distance ladder, 150 feedback, 304, 464, 539–540, 547, 559, 563, 565 distance modulus, 49, 588 by AGNs, 539 distances in cosmology, 186, 188–190, 263 by supernovae, 539 distances of visual binary stars, 53 Fermi bubbles, 97 Dn- relation, 132 Fermi Gamma-Ray Space Telescope, 38, 508 Doppler broadening, 218, 238 fermions, 192, 193 Doppler effect, 48 filaments, 393 Doppler factor, 255 Fingers of God, 400, 401 Doppler favoritism, 255 fireball model. See gamma-ray bursts Doppler shift, 10 flatfield, 292 Doppler width, 11, 238 flatness problem, 207, 208, 387, 458 downsizing, 497, 541 fluid approximation, 343 drop-out technique. See Lyman-break technique flux, 583 dust, 51, 254 free-fall time, 43, 210, 527 Index 619 free-free radiation, 294 star formation, 110 free streaming, 351, 354 stellar orbits, 111 Freeman law, 119, 129 UV-excess, 110 Friedmann equations, 180, 358 green valley, 139 Friedmann–Lemaître model, 15, 16, 180 halos, 122 fundamental plane, 130–132, 541 at high redshift, 460 tilt, 131 high-redshift galaxies FUSE (Far Ultraviolet Spectroscopic Explorer), 35, 427 color-magnitude distribution, 499, 500 demographics, 496–499 interstellar medium, 503 Gaia, 47, 576, 579 metallicity, 503 Galactic center, 7, 89–95 mid-IR luminosity function, 497 black hole, 92–99 morphology, 499 distance, 54, 69–70 optical/NIR luminosity function, 497 Fermi bubbles, 97 size and shape, 499 flares, 95 size evolution, 502 X-ray echos, 95 UV luminosity function, 496 Galactic coordinates, 45–46 Hubble sequence, 103 cylindrical, 46 interacting galaxies, 12 galactic fountain model, 69 IRAS galaxies, 476, 482 Galactic latitude, 45 irregular galaxies, 103, 275 Galactic longitude, 45 late-type galaxies, 8 Galactic plane, 45 LIRG (luminous infrared galaxy), 499 Galactic poles, 45 low surface brightness galaxies (LSBs), 119, 371 galactic winds, 143, 464 luminosity function, 155–158, 523 in Lyman-break galaxies, 464 Lyman-˛ emitters (LAEs), 470 galaxies, 8–9, 101–170 Lyman-break galaxies, 461–466, 471, 475, 521 bimodal color distribution, 105, 140, 541 correlation length, 462 blue cloud, 139 mass function, 158, 560 brightness profile, 108–110, 117 mass-metallicity relation, 142 BzK selection, 469 mass profile, 167 cD galaxies, 108, 292, 300, 322, 330 mean number density, 157 relation to intracluster light, 292 morphological classification, 102 characteristic luminosity L, 102 morphology-density relation, 330–335, 463 chemical evolution, 142–144 morphology of faint galaxies, 471 classification, 102–108, 139–142 narrow-band selection, 470 color-color diagram, 461 polar ring galaxies, 547 color-density relation, 331 post-starburst galaxies, 334 color-magnitude relation, 140, 499 red sequence, 139 color-profile shape relation, 140 S0 galaxies, 103, 552 dark matter fraction, 167 gas and dust, 110 dark matter halo masses, 462 satellite galaxies, 124, 275, 403 distant red galaxies (DRGs), 469, 470 scaling relations, 127–132, 154, 406 dwarf galaxies, 108, 275 spectra, 137–138 E+A galaxies, 334 spheroidal component, 146, 148 early-type galaxies, 8 spiral galaxies, 103, 116–127 elliptical galaxies, 103, 108–116 bars, 103, 117, 126 blue compact dwarfs (BCD’s), 108 bulge, 117, 125, 565 classification, 103, 108 bulges vs. pseudobulges, 118, 538 composition, 110–111 bulge-to-disk ratio, 116 cores and extra light, 109, 546, 548 central surface brightness, 119 counter-rotating disks, 114, 115 color gradient, 125 dark matter, 124 corona, 126, 464 dust lane, 259 dark matter, 122 dwarf ellipticals (dE’s), 108 dust, 125 dwarf spheroidals (dSph’s), 108 dust obscuration and transparency, 125 dynamics, 111–114 early-type spirals, 116 formation, 541–547 gaseous halo, 126–127 gas and dust, 110 gas mass fraction, 124 indicators for complex evolution, 114–116, 546 halo size, 124 interstellar medium, 169 maximum disk model, 123 mass determination, 114 metallicity, 125 mass fundamental plane, 168 normal and barred, 103 mass-to-light ratio, 115 reddening, 125 shape of the mass distribution, 169 rotation curve, 9, 122–124, 128 shells and ripples, 115 spiral structure, 125–126 620 Index
stellar halo, 120, 543 mass profile of dark matter halos, 375–381 stellar populations, 124 mass-sheet degeneracy, 171, 449 thick disk, 120, 543 microlensing effect, 77–88 warps, 119 microlensing magnification pattern, 235 starburst galaxies, 3, 11, 105, 247, 462, 481–484 multiple images, 78, 79, 159, 162–166, 317, 477 sub-millimeter galaxies, 486–493 point-mass lenses, 79–81, 160 AGN fraction, 490 search for clusters of galaxies, 328 correlation length, 490 shear, 323, 376 halo mass, 490 shear correlation function, 419 identification in other wavebands, 488 singular isothermal sphere (SIS) model, 161 mergers, 490 substructure, 385–387 number counts, 488 time delay, 169 redshift distribution, 489, 490 weak lensing effect, 322–329, 419–423, 430 substructure, 381–387, 563 gravitational redshift, 429 suppression of low-mass galaxies, 533 gravitational waves, 388, 548, 549 ULIRG (ultra-luminous infrared galaxy), 25, 105, 254, 482, 483, Great Attractor, 406 499 Great Debate, 101 galaxy evolution, 17, 521–571 Great Observatories Origins Deep Survey (GOODS), 41, 472 numerical simulations, 552–562 Great Wall, 341, 392, 393 overcooling problem, 539 Green Bank Telescope, 20 quasar epoch, 266 green-valley galaxies, 140 semi-analytic models, 562–571 groups of galaxies, 15, 282–283 Galaxy Evolution from Morphology and Spectral Energy Distributions growth factor DC, 345–346, 359, 410, 457 (GEMS) survey, 41, 472 growth of density fluctuations, 342–346 galaxy formation Gunn–Peterson effect, 535 formation of disk galaxies, 536–541 Gunn–Peterson test overview, 522–525 near-zone transmission, 423–424, 529, 534 scale length of disks, 537 galaxy groups, 273, 279, 282–283 compact groups, 283, 291 HII-region, 59 diffuse optical light, 291 hadrons, 193 Galaxy Zoo, 106 harassment in galaxies, 551 GALEX (Galaxy Evolution Explorer), 35, 465, 496 Harrison–Zeldovich fluctuation spectrum, 350, 388, 432, 437 gamma-ray bursts, 516–519 HEAO-1, 36 afterglows, 517 heliocentric velocity, 48 fireball model, 518 helium abundance, 174, 197–198, 452 hypernovae, 518, 519 Herschel blank-field surveys, 499 short- and long-duration bursts, 518 Herschel Lensing Survey (HLS), 477 gauge bosons, 193 Herschel Space Observatory, 26, 486 G-dwarf problem, 143 Hertzsprung–Russell diagram (HRD), 134, 589–591 Gemini telescopes, 31 H.E.S.S. (High Energy Stereoscopic System), 39 General Relativity, 15, 18, 177, 179, 455, 457 Hickson compact groups, 15, 283 Giant Magellan Telescope (GMT), 577 hierarchical structure formation, 327, 361, 365, 382, 486, 521 globular clusters, 66, 125, 148 Higgs mechanism, 194 specific abundance, 125 Higgs particle, 4, 194 gluons, 193 highest-redshift objects, 477 Gran Telescopio Canarias (GTC), 31, 32 high-redshift galaxies, 459–516 gravitational instability, 17, 342–346 size evolution, 546 gravitational lensing, 78, 158–170 high-velocity clouds (HVCs), 68 AGN microlensing, 234–238 in external galaxies, 126 clusters of galaxies as lenses, 317–329, 409, 475 Hipparcos, 47, 48, 576, 590 correlated distortions, 419 Hobby–Eberly Telescope, 31 of cosmic microwave background, 444, 446 Holmberg effect, 275, 387 cosmic shear, 419–423, 444, 449 horizon, 205–206, 352 critical surface mass density, 159, 163 horizon length, 205, 431 deflection angle, 158–159 at matter-radiation equality, 353 differential deflection, 80 horizon problem, 206, 208, 387, 388 Einstein radius, 79, 161, 163, 318 hot dark matter (HDM), 351 Einstein ring, 80, 161, 163 Hubble classification of galaxies, 103, 471, 514 galaxies as lenses, 158–170 Hubble constant H0, 10, 148, 151, 154, 185, 309, 437, 446, 449, 453 galaxy-galaxy lensing, 376–381, 403 scaled Hubble constant h,10 Hubble constant, 169, 309, 449 Hubble Deep Field(s), 31, 41, 470–473 lens equation, 159 galaxy number counts, 471 luminous arcs, 317–322, 460 Hubble diagram, 10, 153, 190, 417 magnification, 80–82, 160, 475–477, 492–493 of supernovae, 152, 415, 417 mass determination, 163, 166 Hubble eXtreme Deep Field (XDF), 472, 473 Index 621
Hubble Key Project, 151, 449 Hercules cluster = Abell 2151, 102 Hubble law, 9, 10, 148, 153, 178, 186 HFLS3, 491, 493 Hubble radius, 177, 342 HXMM01, 490–492 Hubble sequence. See galaxies Hydra A, 14 Hubble Space Telescope (HST), 28, 145, 256, 317, 466, 574 IRAS 13225–3809, 223 Hubble time, 176 IRAS 13349+2438, 247 Hubble Ultradeep Field (HUDF), 41, 471, 472 Leo I, 109 hydrodynamics, 553 M33, 275, 279 hypernovae, 519 M51, 117 hypervelocity stars, 97 M81, 126, 282 M81 group, 279 M82, 3, 282, 484 individual objects M83, 117 ! Centauri, 119 M84, 145, 216 3C 48, 214 M86, 104 3C 75 = NGC 1128, 304 M87, 8, 109, 147, 228, 256, 258 3C120, 227 M94, 117 3C175, 216 MACS J0025.4–1222, 328 3C273, 105, 214, 256, 257 MACS J0647+7015, 479, 480 3C279, 223 MACS J0717.5+3745, 412 3C326, 551 MACS J1206.20847, 4 Abell 68, 475 MCG-6-30-15, 230 Abell 222 & 223, 328 MG 1654+13, 164, 166, 169 Abell 370, 318 MG 2016+112, 387 Abell 383, 14 MS 0735.6+7421, 305 Abell 400, 304 MS 1054–03, 294, 325, 336 Abell 851, 326 MS 1512+36, 476 Abell 1689, 322 NGC 17, 522 Abell 1835, 301 NGC 253, 484 Abell 2218, 319, 320, 477 NGC 454, 522 Abell 2319, 307 NGC 474, 115 Abell 2597, 303 NGC 1068, 212, 214, 252, 253 Abell 3627, 334 NGC 1232, 1 Andromeda galaxy (M31), 101, 120, 150, 275, 278 NGC 1265, 305 Antennae galaxies, 482–484 NGC 1275, 303 Antlia dwarf galaxy, 275, 281 NGC 1365, 117 APM 08279+5255, 477 NGC 1705, 109 Arp 148, 522 NGC 2207 and IC 2163, 12 Arp 220, 12, 482, 492 NGC 2997, 8 Arp 256, 522 NGC 3115, 146 B1938+666, 167 NGC 3190, 119 B2045+265, 386 NGC 3198 (rotation curve), 123 BL Lacertae, 223 NGC 4013, 119 Bullet cluster 1E 0657–56, 305, 324, 327 NGC 4151, 214 Cartwheel galaxy, 106 NGC 4258, 147, 152 cB 58, 464, 475, 476 NGC 4261, 254 Centaurus A = NGC 5128, 109, 257, 259, 543 NGC 4402, 333 Centaurus cluster, 299 NGC 4522, 333 Centaurus group, 279 NGC 4565, 2 CID-42, 551 NGC 4631, 126 CIZA J2242.8+5301, 312 NGC 4650A, 547 Cl 0024+17, 318, 320, 322, 324 NGC 5195, 127 Cl 2244–02, 319, 460 NGC 5548, 223, 240–243, 250 Cl 0053–37, 274 NGC 5728, 244 Cosmic Eye, 476 NGC 5866, 104 Cygnus A, 215 NGC 5907, 104 DLSCL J0916.2+2951, 328 NGC 6050, 102, 522 Dwingeloo 1, 8 NGC 6217, 104 ESO 77-14, 522 NGC 6240, 522, 550 ESO 593-8, 522 NGC 6251, 216 F10214+47, 476 NGC 6670, 522 Fornax dwarf spheroidal, 281 NGC 6786, 522 HCG40, 274 NGC 6822 (Barnard’s Galaxy), 281 HCG62, 304 OJ 287, 550 HCG87, 15 Perseus cluster, 299, 303, 305 Hercules A, 4 Pinwheel galaxy (M83), 122 622 Index
PKS 1127145, 261 Kirchhoff’s law, 584 PKS 2155304, 259, 260 Kormendy relation, 108 PKS 2349, 11 Kuiper Airborne Observatory, 24 PSR J1915+1606, 388 QSO 0957+561, 163, 169 QSO 1422+231, 426 Large Binocular Telescope (LBT), 31, 34, 574 QSO 2237+0305, 163, 166, 237 Large Hadron Collider (LHC), 4, 194, 201, 580 QSO PG1115+080, 163, 165 Large Millimeter Telescope (LMT), 576 QSO ULAS J1120+0641, 481, 534 large-scale structure of the Universe, 388, 392 RXJ 1347–1145, 293 baryon distribution, 426 Sculptor Group, 279 galaxy distribution, 392–408 SDSS J1030+0524, 534 halo model, 378–380, 464 SDSS J1148+5251, 534 non-linear evolution, 357–366 Seyfert’s Sextet, 284 numerical simulations, 361–366, 553–562 SMM J09429+4658, 489 Aquila Comparison Project, 555 SN 1987A, 151 friends-of-friends algorithm, 366 SPT-CL J2106–5844, 308 Hubble Volume Simulation, 363, 365 Stephan’s Quintet (HCG92), 284 inclusion of feedback processes, 553 Tadpole galaxy = Arp 188, 18 Millennium Simulations, 363–369 TN J1338–1942, 339 Virgo Simulation, 364 UDFy-38135539, 478 power spectrum, 350–354, 359, 388, 394–401, 410, 427, 431, 454 UGC 8335, 522 tilt, 388 UGC 9618, 522 Large Synoptic Survey Telescope (LSST), 578 Whirlpool Galaxy (M51), 127 Las Campanas Redshift Survey (LCRS), 393 XMMU J2235.22557, 338 laser guide star, 34 inflation, 207–209, 387–388, 451, 580 Laser Interferometer Space Antenna (LISA), 579 initial mass function (IMF), 85, 133, 166, 167, 510 last-scattering surface, 203 integral field spectroscopy, 34 late-type galaxies, 103 Integral satellite, 38 Leiden-Argentine-Bonn (LAB) survey, 41 integrated Sachs–Wolfe effect (ISW), 430, 432, 440 lenticular galaxies, 103 interactions of galaxies, 268 leptons, 193 interferometry, 19, 20, 574 light cone, 174 intergalactic medium, 423–424, 426, 427, 461, 528, 531 light pollution, 19 intermediate-velocity clouds (IVCs), 68 Limber equation, 405, 458 interstellar medium, 60 linearly extrapolated density fluctuation field, 345 phases, 60 linearly extrapolated power spectrum, 351 intracluster light, 291, 330. See clusters of galaxies LINERs, See active galaxies intrinsic alignments of galaxies, 421 line transitions: allowed, forbidden, semi-forbidden, 238–239, 243 inverse Compton scattering, See Compton scattering Local Group, 15, 273, 275–279 ionization parameter, 240 galaxy content, 275, 384 IRAS (InfraRed Astronomical Satellite), 25, 393, 408, 435, 482, 486 mass estimate, 276–278 IRAS galaxy surveys, 393 local standard of rest (LSR), 71 irregular galaxies. See galaxies look-back time, 190 ISO (Infrared Space Observatory), 25, 482, 486 Lorentz factor, 217, 229 isochrones, 134 Low-Frequency Array (LOFAR), 575 isophote, 103 luminosity isothermal sphere, 161, 287–288, 298, 317 bolometric, 588 IUE (International Ultraviolet Explorer), 35 in a filter band, 588 luminosity classes, 589–591 luminosity distance, 189, 190, 415 James Webb Space Telescope (JWST), 480, 574 luminosity function, 155, 523 Jansky (flux unit), 213 evolution, 497, 498 JCMT (James Clerk Maxwell Telescope), 22 of galaxies, 155–158, 170, 329, 412, 485 Jeans equation, 113–114 of quasars, 264–268, 425 Jeans mass, 529–530 UV LF of high-redshift galaxies, 534 jets, 4, 215, 229, 230, 234, 255–262, 303 luminous arcs. See gravitational lensing beaming, 255–256 luminous red galaxies (LRGs), 397 Doppler favoritism, 255 Lyman-˛ blobs, 495–496 generation and collimation, 256 Lyman-˛ emitters (LAEs), 480 at high frequencies, 256–261 Lyman-˛ forest, 269, 423–428, 461, 465, 531, 536 baryonic acoustic oscillations, 428, 429 damped Ly˛ systems, 269, 424, 493–495 K-correction, 263–264, 487, 488 Lyman-limit systems, 269, 424 Keck telescope, 2, 28, 31, 461, 574 models, 425 Kilo Degree Survey (KiDS), 577 power spectrum, 427 King models, 288, 298 proximity effect, 424 Index 623
as a tool for cosmology, 427–428 structure, 5, 6, 54–70 Lyman-break analogs, 465 thick disk, 55, 58–59 Lyman-break galaxies thin disk, 55, 58 seegalaxies, 461 Millennium Simulation, 363, 567, 569, 571 Lyman-break method, 461–462, 466, 521 mixed dark matter (MDM), 354 Modified Newtonian Dynamics (MOND), 581 molecular clouds, 59 MACHOs, 77, 78, 83–87 moving cluster parallax, 48–49 Madau diagram, 512–514, 559 multi-object spectroscopy, 392–394 Magellanic Clouds, 15, 84, 275, 280 distance, 150 Magellanic Stream, 68, 275 narrow-band photometry, 460, 470, 496 MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov natural telescopes, 475–477, 492–493 Telescopes), 40 Navarro–Frenk–White profile, 367–370 magnification. See gravitational lensing Near Infrared Camera and Multi Object Spectrograph (NICMOS), 30 magnitude, 586–588 neutrinos, 57, 193, 196 absolute magnitude, 587–588 freeze-out, 194–195 apparent magnitude, 586–587 masses, 354, 441, 450, 451 bolometric magnitude, 588 neutrino oscillations, 19, 199 main sequence, 49, 590, 591 radiation component of the Universe, 195, 450 Malmquist bias, 155, 410 Solar neutrino problem, 199 MAMBO (Max-Planck Millimeter Bolometer), 22, 486 Solar neutrinos, 19, 591 mass-energy equivalence, 179 neutron stars, 56, 86 maser, 90, 146 New General Catalog (NGC), 101 mass segregation, 291 New Technology Telescope (NTT), 28 mass spectrum of dark matter halos, 359, 361, 365 non-linear mass-scale, 361 mass-to-light ratio, 59, 135, 137, 278, 291, 412 NVSS (NRAO VLA Sky Survey), 41, 444 in clusters of galaxies, 412 of galaxies, 115, 123, 129, 132 maxBCG group catalog, 380 oblate and prolate, 111 MAXIMA, 436 Olbers’ paradox, 174, 175 MeerKAT, 575 Oort constants, 74, 75 merger tree, 562 open clusters, 59 merging of galaxies, 143, 247, 332, 336, 482, 514, 523, 542–550 optical depth, 50, 584 brightness profiles of merger remnants, 545 optically violently variables (OVV). See active galaxies dry mergers, 332, 544 outflows from galaxies, 143, 539 impact of AGN feedback, 547 overcooling problem, 539, 555 major merger, 543, 565 minor merger, 542, 564 wet mergers, 544 pair production, 506 MERLIN (Multi-Element Radio Linked Interferometer Network), 22 and annihilation, 192, 195–196 mesons, 193 Palomar Observatory Sky Survey (POSS), 40, 279 metallicity, 54, 56, 58, 134, 142 PanSTARRS, 2, 28 metallicity index, 54 parsec, 5, 47 metals, 54 particle-mesh (PM) method, 363 microlensing. See gravitational lensing particle-particle particle-mesh (P3M) method, 363 Milky Way, 45–99 passive evolution of a stellar population, 138 annihilation radiation, 64 Pauli exclusion principle, 193 bar, 84 peak-background split, 374 bulge, 64 peculiar motion, 150, 153 center (see Galactic center) peculiar velocity, 71, 344, 346–347, 399–401, 406–408 chemical composition, 56 period-luminosity relation, 53, 54, 150, 151 dark halo, 6, 83, 85 perturbation theory, 343–346 disk, 55–61 photometric redshift, 466–468, 471, 480 distribution of dust, 60 catastrophic outliers, 468 gamma radiation, 64 Pico Veleta telescope, 22 gas, 55, 59–61, 271 Pierre Auger Observatory, 63 gaseous halo, 67–69 Planck function, 187, 585 halo, 66–70 Planck mass, 200 hypervelocity stars, 97 Planck satellite, 24, 203, 441–447 infalling gas, 67 planetary nebulae, 58, 593 kinematics, 70–77 as distance indicators, 154 magnetic field, 61–62 Plateau de Bure interferometer, 487 multi-wavelength view, 7 point-spread function, 32, 420 rotation curve, 6, 43, 73–77 Poisson equation, 344 stellar streams, 66 in comoving coordinates, 345 624 Index polarization, 61 Saha equation, 202 population III stars, 531, 540 Salpeter initial mass function. See initial mass function (IMF) population synthesis, 133–138, 541 satellite galaxies. See galaxies, 384, 464 power spectrum, 350, 354, 359, 454 scale factor. See Universe of galaxies, 396, 398 scale-height of the Galactic disk, 55 normalization, 395, 410, 422, 429 Schechter luminosity function, 155, 329, 412, 496 shape parameter , 395 Schmidt–Kennicutt law of star formation, 120–121, 538 Press–Schechter model, 359–361, 365, 529, 563 SCUBA (Sub-millimeter Common-User Bolometer Array), 22, 486 pressure of radiation, 181 SCUBA-2, 22 primordial nucleosynthesis (BBN), 16, 196–198, 391, 441, 450, 452 SDSS Quasar Lens Search (SQLS), 162 baryon density in the Universe, 413, 454 secondary distance indicators, 153–155, 406 primordial spectral index ns, 441 seeing, 19, 28, 47 proper motion, 48, 92 self-shielding, 493 proto-cluster, 463, 486 semi-analytic model of galaxy evolution, 562–571 proximity effect, 424 sensitivity of telescopes, 19 pseudobulges, 118 Sérsic brightness profile, 139 pulsating stars, 53 Sérsic index, 139, 140 service mode observing, 33 Seyfert galaxies. See Active galaxies QSOs. See active galaxies SgrA. See Galactic center quarks, 193 shape parameter , 353, 395, 405, 414, 429 quasars. See active galaxies shells and ripples, 115 shock fronts, 63, 217, 257, 305, 311 shock heating, 525 radial velocity, 48 Silk damping, 430, 432 radiation force, 230 singular isothermal sphere (SIS), 161, 288, 317 radiative transfer equation, 583–584 Sloan Digital Sky Survey (SDSS), 41, 139, 264, 330, 343, 394, 405 RadioAstron, 22 Sloan Great Wall, 341, 343 radio galaxies. See active galaxies Sloan Lens Advanced Camera for Surveys (SLACS), 162 ram-pressure stripping, 332–334, 552 smooth particle hydrodynamics (SPH), 553, 554 random field, 347–348 SOFIA (Stratospheric Observatory for Infrared Astronomy), ergodicity, 349 27 Gaussian random field, 350 softening length, 363 realization, 348 sound horizon, 355, 398, 432 Rayleigh–Jeans approximation, 487, 585 sound velocity in photon-baryon fluid, 355 reaction rate, 195 sound waves, 355 recombination, 16, 201–203, 528 source counts in an Euclidean universe, 176 two-photon decay, 202 source function, 584 reddening vector, 52 South Pole Telescope (SPT), 24, 25, 441, 494 red cluster sequence (RCS), 283, 336, 468, 541 Space Telescope Imaging Spectrograph (STIS), 30 red giants, 590, 592, 593 specific energy density of a radiation field, 583 red-sequence galaxies, 140 specific intensity, 187, 583 redshift, 10, 174 spectral classes, 589–591 cosmological redshift, 186–188, 227 spectral resolution, 19 desert, 521 spectroscopic distance, 52 relation to scale factor, 186 spherical accretion, 226 space, 400 spherical collapse model, 357–359, 525 redshift space distortions, 399–401 spin parameter, 373, 536 anisotropy of the correlation function, 400, 401 spiral arms, 117, 125 redshift surveys of galaxies, 392–408 as density waves, 125 reionization, 203, 424, 430, 439, 440, 478, 528–536 spiral galaxies. See galaxies helium reionization, 533 Spitzer Space Telescope, 26, 483, 497 observational probes, 534–536 Square Kilometer Array (SKA), 575 UV-slope of high-redshift galaxies, 535 standard candles, 58, 152–153 relaxation time-scale, 112–113, 290 starburst-AGN connection, 514 reverberation mapping, 239–243, 249 starburst galaxies. See galaxies right ascension, 46 star formation, 59, 486, 511–512, 524, 529 ROSAT (ROentgen SATellite), 36, 126, 310, 314, 410, 508 and color of galaxies, 135, 336, 466, 521 ROSAT All-Sky Survey (RASS), 36, 310, 414 comparison of indicators, 512 rotational flattening, 111 cosmic history, 510–514, 559 rotation measure, 62 different modes, 514 RR Lyrae stars, 53, 69 efficiency, 523 runaway stars, 99 feedback processes, 539–540 initial mass function (IMF), 133 Sachs–Wolfe effect, 429, 431 Madau diagram, 512–515 Sagittarius dwarf galaxy, 6, 278, 543 quiescent star formation, 514 Index 625
rate, 105, 133, 247, 425, 465 trigonometric parallax, 47–48 Schmidt–Kennicutt law, 120–121 of Cepheids, 151 star-formation burst, 491 Tully–Fisher relation, 128–130, 562, 568 star-formation rate (SFR), 510–512 baryonic, 129 star-formation rate density, 510, 559 Two-Degree-Field Survey, 264, 341, 394, 396, 401 Stefan–Boltzmann law, 182, 586 Two Micron All Sky Survey (2MASS), 41, 316, 408 stellar evolution, 541 stellar mass estimate, 137 stellar mass function, 534 ultra-luminous compact X-ray sources (ULXs), 483, 484 stellar populations, 55, 56 ultra-luminous infrared galaxies (ULIRGs). See galaxies stellar streams, 66, 120 Universe strangulation in galaxies, 552 age, 176, 185 strong interaction, 193 baryon asymmetry, 200 Subaru telescope, 31, 574 baryon-to-photon ratio, 196, 198, 202 Sunyaev–Zeldovich effect, 306–309, 430, 434, 442, 450 critical density, 179, 206, 367 Compton-y parameter, 307 as a function of redshift, 206 distance determination, 309 curvature, 441, 450, 454 Hubble constant, 309 density, 15, 17, 43, 454 integrated Compton-y parameter, 307 density fluctuations, 17, 341–388 kinetic SZ effect, 309 density parameter, 185, 198, 206, 410, 412, 413, 422 superclusters, 341, 412 Einstein–de Sitter model, 17, 185, 190, 346, 359, 360 superluminal motion, 227–229, 252, 255 expansion, 9, 177, 178, 194, 344 supernovae, 56, 593 homogeneity scale, 342 classification, 56 scale factor, 177, 184 core-collapse supernovae, 56 standard model, 4, 204–207, 384, 391, 438 as distance indicators, 58, 152, 414–418, 449 thermal history, 16, 192–204 evolutionary effects, 417 at high redshift, 414–415 metal enrichment of the ISM, 56–58, 65, 531 vacuum energy. See dark energy SN 1987A, 57, 150 variation of physical constants, 451 Type Ia, 58 velocity dispersion, 55 single- vs. double-degenerate progenitor, 58 in clusters of galaxies, 287, 288 supersymmetry, 200 in galaxies, 111, 147, 161 surface brightness fluctuations, 153 Very Large Array (VLA), 20, 256, 488 surface gravity, 590 Very Large Telescope (VLT), 31, 574 surveys, 40–41 Very Long Baseline Array (VLBA), 22 Suzaku, 37 Very Long Baseline Interferometry (VLBI), 22 SWIFT, 518 VIMOS VLT Deep Survey (VVDS), 394 synchrotron radiation, 62, 216–218, 234, 257 violent relaxation, 290, 358 cooling time of electrons, 218, 257 Virgo cluster of galaxies, 13, 150, 151, 279 polarization, 217 intracluster light, 292 spectral shape, 217–218 virial mass, 312 synchrotron self-absorption, 217 virial radius, 312, 367 synchrotron self-Compton radiation, 258, 259 virial temperature, 525, 530 virial theorem, 13, 225, 289 virial velocity, 367 tangential velocity, 48 virtual observatory, 42, 579 tangent point method, 75 VISTA, 577 TeV-astronomy, 39 VLT Survey Telescope (VST), 577 thermal radiation, 585 voids, 12, 341, 393 Thirty Meter Telescope (TMT), 577 Voigt profile, 269 Thomson cross section, 232 Thomson scattering, 231, 430, 433 three-body dynamical system, 98 W- and Z-boson, 193 throughput, 19 warm dark matter, 354, 428 tidal disruption, 66 warm-hot intergalactic medium, 427 tidal disruption event (TDE), 262–263 wave number, 350 tidal streams, 278 weak interaction, 193, 195 tidal stripping, 66 weak lensing effect. See gravitational lensing tidal tails, 482, 543 wedge diagram, 392, 399 tidal torque, 372 white dwarfs, 58, 85, 417, 590, 593 time dilation, 417 Wide Field and Planetary Camera (WFPC2), 30 tip of red giant branch, 152 Wide Field Camera 3 (WFC3), 30, 323, 472 transfer function, 352–354, 357, 395 width of a spectral line, 218 qualitative behavior, 352–354 Wien approximation, 585 triaxial ellipsoid, 111 Wien’s law, 585 626 Index
WIMPs, 195, 198–201, 580 X-ray background, 203 direct detection experiments, 200 X-ray binaries, 226, 482 indirect astrophysical signatures, 201 X-ray source counts, 474 WISE (Wide-field Infrared Survey Explorer), 27 WMAP (Wilkinson Microwave Anisotropy Probe), 24, 203, 438–441, 445–447 yield, 143 Wolter telescope, 36
Zeeman effect, 61 X-factor, 60 zero-age main sequence (ZAMS), 591 XMM-Newton, 36, 246, 299 zodiacal light, 60 X-ray absorption, 247 Zone of Avoidance, 46, 273, 407