5. GRAVITATIONAL INTERACTION: STRUCTURE OF THE MACROCOSMOS


  5.1 Structural Elements of the Universe

    The structure of the macrocosmos is manifested on many different scales, ranging from the universe on the largest scale, down to galaxies, stars, and planets. Only objects of the microcosmos, such as quarks and leptons, may be devoid of further substructure (However, a recent experiment is suggesting that quarks and gluons may be composed of more fundamental particles; see Wilczek, 1996.) Most scales are determined to an order of magnitude by a few physical constants. In particular, the mass scale and length scale (in units of the proton mass mp and the Bohr radius a0) of structures down to the planets can be expressed in terms of the electromagnetic fine structure constant alphae , Eq.(4), the gravitational fine structure constant alphag , Eq.(3), and the electron-to-proton mass ratio (mp round 1837me).

    The following considerations are based on order of magnitude arguments, factors of order unity (like pi) being neglected. These results have been published in the astrophysical literature at several occasions and are collected here for easy reference (a detailed discussion is contained in Carr and Rees, 1979; Barrow and Tipler, 1988).

5.1.1 The Planck scales

    The only quantities of dimensions mass and length which can be constructed from G, h, and c are the Planck scales:

M_sub_Planck proportional to (G/hc)^(-1/2) proportional to 10^(-5) g, R_sub_Planck proportional to (G/hc^3)^(1/2) proportional to 10^(-33) cm . (9)

    Using the gravitational fine structure constant, eq.(3), these scales can be expressed as

M_sub_Planck proportional to (alpha_sub_G)^(-1/2) * m_sub_p, R_sub_Planck proportional to (alpha_sub_G)^(1/2) * r_sub_p . (10)

    is much larger than mp but RPlanck is much smaller than rp (rp being the size of a proton that can be taken to be the Compton wavelength associated with its rest mass, rp proportional h/mpc proportional 10-13 cm; accordingly, the corresponding timescale is tp proportional rp/c proportional 10-23 s). The Planck length is the scale on which quantum gravitational fluctuations in the metric become of the order of unity, so the concept of space breaks down at such small scales. MPlanck can be interpreted as the mass of a black hole of radius RPlanck. Space may be thought of as being filled with virtual black holes of this size. Such "instantons" may play an important role in quantum gravity theory.

5.1.2 The Universe

    In the simplest Friedmann cosmological model, the age of the universe t0, is of the order of H0-1 where H0 is the Hubble parameter (this relation fails only if the universe is closed and near its maximum expansion). Since H0 proportional 50 km s-1 Mpc-1, this implies t0 proportional 1010 yr, a conclusion which is supported by several independent arguments. The associated horizon size (the distance travelled by light since the birth of the universe) satisfies the approximate relationship

c * t_sub_0 proportional to (alpha_sub_g)^(-1) * [h/ (m_sub_e * c)] proportional to (alpha_sub_e / alpha_sub_g) * a_sub_0 , (11)

    where a0 denotes the radius of the lowest energy electron orbit of a hydrogen atom, a0 proportional h2/mee2 proportional 10-8 cm proportional 1 Bohr. The ratio of the size of the observable universe to the size of an atom is comparable with the ratio of the electrical (or nuclear) and gravitational forces between elementary particles. There is no explanation for this well known coincidence within conventional physics, but Dirac (1937, 1938) has conjectured that alpha g is always given by

alpha_sub_g proportional to h / (m_sub_e * c^2 * t) proportional to (t / t_sub_e)^(-1) . (12)

    Assuming that h, c, and me are constant in time, this requires that G decreases as t-1, so Dirac invokes Equation (12) as the basis for a new cosmology (Barrow, 1996). Such a variation of G is inconsistent with current observations. The total mass associated with the observable universe (the mass within the horizon volume) is proportional rho0c3t03, where rho0 is the present matter density, given by the Friedmann equation,

rho_sub_0 = {(3 * [H_sub_0]^2) / (8 * Pi * G)} + {(K * c^2) / (16 * Pi * G)} . (13)

    Here K is the scalar curvature of the universe. Providing the K term is smaller than the others, one deduces that the mass of the universe is

M_sub_u proportional to (c^3 * [t_sub_0]^3 * G^(-1) * [H_sub_0]^2) proportional to [(c^3 * t_sub_0) / (G)] proportional to ([alpha_sub_g]^(-2) * [m_sub_p / m_sub_e] * m_sub_p) . (14)

    The fact that the number of protons in the universe is of the order of alphag-2 is thus explained providing one can justify neglecting the Kterm in Equation (13). It has been argued that K must always be zero by appealing to Mach's principle but, apart from this, there may be reasons for expecting that the K term is small. If K is negative, galaxies could not have condensed out from the general expansion unless (-K) were less than Grho/c2 at their formation epoch. Otherwise, their gravitational binding energy would not have been large enough to halt their expansion. The term (-K) may exceed Grho/c2 at the present epoch, but not by a large factor. If K is positive, it must be < Grho/c2, otherwise the whole universe would have recollapsed before t proportional_equal tMS where tMS is the lifetime of a typical main-sequence star, say, the Sun. Relationships (11) and (14) also mean that the universe has an optical depth of the order of unity to electron scattering.

 

  5.1.3 Galaxies

    It is not certain how galaxies form, so any estimate of their scale is very model dependent (Rees and Ostriker, 1977; Silk, 1977; Sciama, 1953). One can assume that galaxies originate from overdense regions in the gaseous primordial material, and that they have a mass M and radius RB when they become bound. After binding, motions in the protogalaxy randomize and equilibrate in the gravitational field of the galaxy at a radius proportional RB/2. Thereafter, they will deflate on a cooling timescale, with a virial temperature

T proportional to (G * M * m_sub_p) / (k * R) . (14)

    Provided kT exceeds one Rydberg the dominant cooling mechanism is bremsstrahlung and the associated cooling timescale is

t_sub_cool proportional_equal to {([m_sub_e]^2 * c^3) / (alpha_sub_e * e^4 * n)} * {(k * T) / (m_sub_e * c^2)} ^ (1/2) . (16)

    The free-fall timescale is

t_sub_ff proportional to (G * M / [R^3])^(-1/2) , (17)

    and this exceeds tcool when R falls below a mass-independent value

R_sub_g proportional to [alpha_sub_e]^4 * [alpha_sub_g]^(-1) * (m_sub_p / m_sub_e)^(1/2) * a_sub_0 , (18)

    which, from a more precise calculation, is 75 kpc. Until a massive cloud gets within this radius it will contract quasi-statically and cannot fragment into stars. This argument applies only if the mass is so high that kTvirial > alphae2mec2 at the "magic radius" Rg; that is,

M >= M_sub_g = [alpha_sub_g]^(-3/2) * [alpha_sub_e]^5 * (m_sub_p / m_sub_e)^(1/2) proportional_equal to M_sub_Sun . (19)

    Gas clouds of mass < Mg cool more efficiently owing to recombination, and can never be pressure supported. Thus, Mg is a characteristic maximum galactic mass. Primordial clouds of mass < Mg are inhibited from fragmentation and may remain as hot pressure-supported clouds. This type of argument can be elaborated and made more realistic (White and Rees, 1978; Rees and Ostriker, 1977; Silk, 1977; Sciama, 1953); but one still obtains a mass proportional Mg above which any fluctuations are likely to remain amorphous and gaseous, and which may thus relate to the mass-scale of galaxies. The quantities Mg and Rg may thus characterise the mass and radius of a galaxy. These estimates is the least certain. The properties of galaxies may be a consequence of irregularities imprinted in the universe by processes at early epochs.

 

  5.1.4 Stars

    The virial theorem implies that the gravitational binding energy of a star must be of the order of its internal energy. Its internal energy comprises the kinetic energy per particle (radiation pressure being assumed negligible for the moment) and the degeneracy energy per particle. The degeneracy energy will be associated primarily with the Fermi-momentum of the free electrons, p proportional h/d, where d is their average separation. Provided the electrons are non-relativistic, the degeneracy energy is p2/2me, so the virial theorem implies (Dyson, 1972; Weisskopf, 1975; Haubold and Mathai, 1994)

kT + [(h^2) / (2 * m_sub_e * d^2)] proportional to [(GM * m_sub_p) / R] proportional to (N / N_sub_0)^(2/3) * (hc / d) . (20)

    Here N is the number of protons in the star, N0 3 equal dashes alphag-3/2, and R proportional N1/3d is its radius. As a cloud collapses under gravity, Equation (20) implies that, for large d, T increases as d-1. For small d, however, T will reach a maximum

k * T_sub_max proportional to (N / N_sub_0)^(4/3) * m_sub_e * c^2 , (21)

    and then decrease, reaching zero when d is

d_min proportional to (N / N_sub_0)^(-2/3) * r_sub_e , (22)

    where re is the size of an electron, re proportional h/mec proportional 10-10 cm; accordingly the electron timescale is defined as te = re/c proportional 10-20s. A collapsing cloud becomes a star only if Tmax is high enough for nuclear reactions to occur, that is kTmax > qmec2 where q depends on the strong and electromagnetic interaction constants and is proportional 10-2. From Equation (21), one therefore needs N > 0.1N0. Once a star has ignited, further collapse will be postponed until it has burnt all its nuclear fuel. An upper limit to the mass of a star derives from the requirement that it should not be radiation-pressure dominated. Such a star would be unstable to pulsations which would probably result in its disruption. Using the virial theorem (that is, Equation (20) with the degeneracy term assumed negligible) to relate a star's temperature T to its mass M proportional Nmp and radius R, the ratio of radiation pressure to matter pressure can be shown to be

(P_sub_rad / P_sub_mat) proportional to [(a * T^4 * R^3) / (N * k * T)] proportional to [N / N_sub_0]^2 , (23)

    so the upper limit to the mass of a star is also proportional N0mp. A more careful calculation shows that there is an extra numerical factor of the order of 10, so one expects all main-sequence stars to lie in the range 0.1 < N/N0 < 10 observed. Only assemblages of 1056 - 1058 particles can turn into stable main sequence stars with hydrogen-burning cores. Less massive bodies held together by their own gravity can be supported by electron "exclusion principle" forces at lower temperatures (they would not get hot enough to undergo nuclear fusion unless squeezed by an external pressure); heavier bodies are fragile and unstable owing to radiation pressure effects. The central temperature, T, adjusts itself so that the nuclear energy generation rate balances the luminosity, the radiant energy content divided by the photon leakage time,

L proportional_equal to a * c * T^4 * R^4 / (kappa * M) , (24)

    where kappa is the opacity. The appropriate kappa decreases as M increases (electron scattering being the dominant opacity for upper main-sequence stars) but the energy generation increases so steeply with T that M/R depends only weakly on M.

 

  5.1.5 White Dwarfs and Neutron Stars

    When a star has burnt all its nuclear fuel, it will continue to collapse according to Equation (20) and, providing it is not too large, it will end up as a cold electron-degeneracy supported "white dwarf" with the radius R proportional_alpha M-1/3 indicated by Equation (22). However, when M gets so large that kTmax proportional mec2, the electrons will end up relativistic, with the degeneracy energy being pc rather than p2/2me. Thus the degeneracy term in Equation (20) acts as d-1 instead of d-2, and consequently there is no T = 0 equilibrium state. From Equation (21) this happens if M exceeds the mass

M_sub_c proportional to [alpha_sub_g]^(3/2) * m_sub_p proportional to M_sub_Sun , (25)

    which characterises stars in general. A more precise expression for this critical value of M (Chandrasekhar mass) is 5.6mu2MSun, where mu is the number of free electrons per nucleon (Chandrasekhar, 1935). For a star which has burned all the way up to iron, mu round 1/2, and the limiting Chandrasekhar mass, taking into account the onset of inverse ß-decay when electrons get relativistic, is 1.25 MSun. Stars bigger than Mc will collapse beyond the white dwarf density but may manage to shed some of their mass in a supernova explosion. The remnant core will comprise mainly neutrons, the electrons having been squeezed onto the protons through the reaction p + e- -> n + v, and this core, if small enough, may be supported by neutron-degeneracy pressure. The limiting mass for a neutron star is more difficult to calculate than that for a white dwarf because of strong interaction effects and because, from Equation. (21), with me -> mp, the particles which dominate the neutron star's mass are relativistic. The maximum mass is still of the order of MSun, however, and a neutron star bigger than this must collapse to a black hole. The maximum mass lies close to the intercept of a black hole line, given by Equation (26), and the nuclear density line rho proportional mp/rp3. The intricacies of the line which bridges white dwarf and neutron star regimes reflect the effects of gradual neutronisation and strong interactions. Stars on this bridge would be unstable and so are not of immediate physical interest.

    The above order-of magnitude arguments show why the effects of radiation pressure and relativistic degeneracy both become important for masses > alphag-3/2mp. Note also that general relativity is unimportant for white dwarfs because the binding energy per unit mass is only proportional (me/mp) of c2 at the Chandrasekhar limit.

5.1.6 Black holes

    The radius of a spherically symmetrical black hole of mass M is

R = 2GM / c^2 proportional to alpha_sub_g * (M / m_sub_p) * r_sub_p . (26)

    This is the radius of the event horizon, the region from within which nothing can escape, at least, classically. Black holes larger than MSun may form from the collapse of stars or dense star clusters. Smaller holes require much greater compression for their formation than could arise in the present epoch, but they might have been produced in the first instants after the Big Bang when the required compression could have occurred naturally. Such "primordial" black holes could have any mass down to the Planck mass. In fact, Hawking (1974) has shown that small black holes are not black at all; because of quantum effects they emit particles like a black body of temperature given by

k * theta proportional to [(h * c^3) / (GM)] proportional to [alpha_sub_g]^(-1) * (M / m_sub_p)^(-1) * m_sub_p * c^2 . (27)

    This means that a hole of mass M will evaporate completely in a time

t_sub_evap proportional to [alpha_sub_g]^2 * (M / m_sub_p)^3 * t_sub_p * [N(theta)]^(-1) . (28)

    N(theta) is the number of species contributing to the thermal radiation: For ktheta < Mec2 these include only photons, neutrinos, and gravitons; but at higher temperatures other species may contribute. The evaporation terminates in a violent explosion. For a solar mass hole, this quantum radiance is negligible: theta is only 10-7 K and tevap proportional 1064 yr. But for small holes it is very important. A Planck mass hole has a temperature of 1032 K and only survives for a time proportional RPlanck/c proportional 10-43 s. Those holes which are terminating their evaporation in the present epoch are particularly interesting. As the age of the universe is t0 proportional alphag-1te, the mass of such holes would be

M_sub_h proportional to [alpha_sub_g]^(-2/3) * (t_sub_0 / r_sub_p)^(1/3) * m_sub_p proportional to [alpha_sub_g]^(-1) * m_sub_p proportional to 10^15 g . (29)

    and , from Equation (26), their radius would be rp. The corresponding temperature is proportional 10 MeV: Low enough to eliminate any uncertainty in N(theta) due to species of exotic heavy particles.

5.2 Evolution of the Universe

    The origin of the universe is governed by laws of physics which are still unknown at the time of writing the Encyclopedia of Applied Physics (see Figure 6).

 


 

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Figure 6.The Big Bang: From the origin of the universe to its present epoch (Smoot and Davidson, 1995).

 


 

    At t = 10-43 s, T = 1032 K: The strong, weak, and electromagnetic forces may appear as unified into one indistinguishable force. This period is often referred to as the Grand Unification epoch. During this epoch, there may have been a very rapid, accelerating expansion of the universe called "inflation". The inflation made the universe very large and flat, but also produced ripples in the space-time it was expanding.

    At t = 10-34 s, T = 1027 K: The strong force becomes distinct from the weak and electromagnetic forces. The universe is a plasma of quarks, electrons, and other particles. Inflation ends and the expanding universe coasts, gradually slowing its expansion under the pull of gravity.

    At t = 10-10 s, T = 1015 K: The electromagnetic and weak forces separate (see Figure 7). An excess of one part in a billion of matter over antimatter has developed. Quarks are able to merge to form protons and neutrons. Particles have acquired substance.

 


 

Figure 7. Cosmology has been actively investigating the consequences of a new extension of the theory of matter within the evolution of the universe, in which the electricity, magnetism, weak force, strong force, and gravity are all unified at sufficiently high temperatures.

 


 

    At t = 1 s, T = 1010 K: Neutrinos decouple and the electrons and positrons annihilate, leaving residual electrons but predominantly the cosmic background radiation as the main active constituent of the universe.

    At t = 3 min, T = 109 K: Protons and neutrons are able to bind together to form nuclei since their binding energy is now greater than the cosmic background radiation energy. A rapid synthesis of light nuclei occurs - first deuterium (D), then heavier elements, primarily helium (3He, 4He) but up to lithium nuclei 7Li (Tytler, Fan, and Burles, 1996; Gloeckler and Geiss, 1996). About 75 percent of the nuclei are hydrogen and 25 percent are helium; only a tiny amount are other elements. The heavier elements are later formed by nuclear burning stars.

    At t = 3 cross 105 yr, T = 3 cross 103 K: Matter and the cosmic background radiation decouple as electrons bind with nuclei to produce neutral atoms. The universe becomes transparent to the cosmic background radiation, making it possible for the Cosmic Background Explorer satellite (COBE) to map this epoch of last scattering (see Figure 8).

 


 

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Figure 8. Three full-sky maps made by the COBE satellite DMR instrument show (Smoot and Davidson, 1995).

Top: The dipole anisotropy caused by the Earth's motion relative to the cosmic background radiation (hotter in the direction we are going, cooler in the direction we are leaving).

Center: The dipole-removed sky showing the emission from the plane of the galaxy - the horizontal red strip and the large-scale ripples in space-time.

Bottom: A map of the wrinkles in time.

 


 

    At t = 109 yr, T = 18 K: Clusters of matter have formed from the primordial ripples to form quasars, primordial stars, and protogalaxies. In the interior of stars, the burning of the primordial hydrogen and helium nuclei synthesizes heavier nuclei such as carbon, nitrogen, oxygen, and iron. These are dispersed by stellar winds and supernova explosions, making new stars, planets, and life possible.

    At t = 15 cross 109 yr, T = 3 K: The present epoch is reached (see Figure 9). Five billion years earlier, the solar system condensed from the remnants of earlier stars. Chemical processes have linked atoms together to form molecules and then solids and liquids. Man has emerged from the dust of stars to contemplate the universe around him.

 


 

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Figure 9. The Big Bang, with inflation producing space and the ripples in space-time mapped by COBE, and eventually evolving to become stars and galaxies and clusters of them (Smoot and Davidson, 1995).

 


  5.3 Large-Scale Distribution of Matter

    The nearest large galaxy to the Milky Way is the "Andromeda galaxy" which is about 670 kpc away. Its mass is proportional3 cross 1011MSun (or perhaps as much as 1.5 cross 1011MSun if it has a massive halo) and it has a size of proportional50 kpc. Studies show that it is a spiral galaxy. The Andromeda galaxy and the Milky Way lie nearly in each other's planes, but their spins are opposite. It may be noted that galaxies are packed in the universe in a manner very different from the way the stars are distributed inside a galaxy: The distance from the Milky Way to the nearest large galaxy is only 20 galactic diameters while the distance from the Sun to the nearest star is thirty million times the diameter of such individual stars. There is some evidence to suggest that the Andromeda galaxy and the Milky Way are gravitationally bound to each other. They have a relative velocity - towards each other - of about 300 km s-1. These two are only the two largest members of a group of more than 30 galaxies all of which together constitute what is known as the "Local Group." The entire Local Group is irregular in shape and can be contained within a spherical volume of proportional2 Mpc in radius. This kind of clustering of galaxies into groups is typical in the distribution of the galaxies in the universe. A study within a size of about 20 Mpc from the Milky Way shows that only (10 - 20)% of the galaxies do not belong to any group; they are isolated galaxies, called "field" galaxies. Groups may typically consist of up to 100 galaxies; a system with more than 100 galaxies is conventionally called a "cluster." The sizes of groups range from a few hundred kpc to 2 Mpc. Clusters have a size of typically a few Mpc. Just like galaxies, one may approximate clusters and groups as gravitationally bound systems of effectively point particles. The large gravitational potential energy is counterbalanced by the large kinetic energy of random motion in the systems. The line of sight velocity dispersion in groups is typically 200 km s-1, while that in clusters can be nearly 1000 km s-1.

    There are several similarities between clusters of galaxies and stars in an elliptical galaxy. For example, the radial distribution of galaxies in a cluster can be adequately fitted by the R1/4 law with an effective radius Re round (1 - 2)h-1 Mpc (0.5 smaller equal h smaller equal 1 related to the Hubble constant). About 10% of all galaxies are members of large clusters. In addition to galaxies, clusters also contain very hot intracluster gas at temperatures 107 - 108 K. The two large clusters nearest to the Milky Way are the Virgo cluster and the Coma cluster. The Virgo cluster, located proportional17 Mpc away, has a diameter of about 3 Mpc and contains more than 2500 galaxies. This is a prominent irregular cluster and does not exhibit a central condensation or a discernible shape. Virgo has an intracluster X-ray emitting gas with a temperature of proportional108 K, which has at least ten times the visible mass of the cluster. The Coma cluster, on the other hand, is almost spherically symmetric with a marked central condensation. It has an overall size of proportional3 Mpc while its central core is proportional600 kpc in size. The core is populated with elliptical and spheroidal galaxies with a density nearly thirty times larger than the Local Group. These values are typical for large clusters. Coma is located at a distance of proportional80 Mpc and contains more than 1000 galaxies. The distribution of galaxies around the Local Group has been studied extensively. It turns out that most of the galaxies nearby lie predominantly in a plane - called the super-galactic plane - which is approximately perpendicular to the plane of the Milky Way galaxy. The dense set of galaxies in this plane is called the Local Supercluster (radius proportional37 Mpc), and the Virgo cluster is nearly at the centre of this highly flattened disc-like system. The term "supercluster" is just used to denote structures bigger than clusters. Broadly speaking, the Local Supercluster consists of three components: About 20% of the brightest galaxies, forming the core, is the Virgo cluster itself; another 40% of galaxies lie in a flat disc with two extended, disjoint groups of galaxies; the remaining 40% is confined to a small number of groups scattered around. Nearly 80% of all matter in the Local Supercluster lies in a plane. The Local Supercluster is clearly expanding. Studies of distant galaxies show that there are many superclusters in our universe separated by large voids. The distribution of matter seems to be reasonably uniform when observed at scales bigger than about 100h-1. For comparison, note that the size of the observed universe is about 3000h-1. Thus, one may treat the matter distribution in the universe to be homogeneous while dealing with the phenomena at scales larger than 100 Mpc. The standard cosmological model is based on this assumption of "large scale" homogeneity (see Figure 10 and Table 4; Geller and Huchra, 1989).

 


 

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Figure 10. A map of the nearby universe toward the north and south poles of the Milky Way. Each of the 9325 points in the image represents a galaxy similar to the Milky Way. The arcs which form the boundaries of the two wedge-like portions of the map are all at a distance of about 400 million light years from the Sun (Earth). The dark regions to the east and west are obscured by the plane of the Milky Way. The map shows that galaxies are arranged in patterns on an enormous scale. The Great Wall, a sheet containing thousands of galaxies, stretches nearly horizontally across the entire northern portion of the survey. A similar Southern Wall runs diagonally across the southern region. These walls delineate enormous dark voids where there are few if any galaxies. The voids are often 150 million light years in diameter. The patterns in the north and south are similar. These large patterns are a tough challenge for attempts to model the development of structure in the universe. The curved boundaries are lines of constant declination (Galactic latitude): in the north they are at 8.5o and 44.5o (the wedge subtends an angle of 36o in the narrow direction). It runs from 8 to 17 hours in right ascension (longitude) or of order 120o. In the south the declination runs from 0o to -40o and the right ascension runs from 20.8 to 4 hours. By permission of Margaret J. Geller, John P. Huchra, Luis A. N. da Costa, and Emilio E. Falco, Smithsonian Astrophysical Observatory © 1994.

 


 
Table 4. Structural elements of the macrocosmos.

Object Mass (g) Radius (cm)

Jupiter 2 x 1030 6 x 109
Sun 2 x 1033 7 x 1010
Red giant (2 - 6) x 1034 1014
White dwarf 2 x 1033 108
Neutron star 3 x 1033 106
Open cluster 5 x 1035 3 x 1019
Globular cluster 1.2 x 1039 1.5 x 1020
Spiral 2 (x 1044 - x 1045) (6 - 15) x 1022
Elliptical 2 (x 1043 - x 1045) (1.5 - 3) x 1023
Group 4 x 1046 3 x 1024
Cluster 2 x 1048 1.2 x 1025

 


  5.4 Morphology of Galaxies

    Galaxies range widely in their sizes, shapes, and masses; nevertheless, one may talk of a typical galaxy as something made out of proportional1011 stars. Taking the mass of a star to be that of the Sun, the luminous mass in a galaxy is proportional1011MSun round 2 cross 1044 g.

    This mass is distributed in a region with a size of proportional 20 kpc. Even though most galaxies have a mass of proportional(1010 - 1012)MSun and a size of proportional10 - 30 kpc, there are several exceptions at both ends of this spectrum. For example, dwarf galaxies" have masses in the range proportional(105 - 107)MSun and radii of proportional1 - 3 kpc. There are also some "giant galaxies" with masses as high as 1013MSun. Galaxies exhibit a wide variety in their shapes as well and are usually classified according to their morphology (Hubble's classification of galaxies, for example). One may divide them into "ellipticals" and "discs". Ellipticals are smooth, featureless, distributions of stars, ranging in mass from 108MSun to 1013MSun. The proportion of elliptical galaxies in a region depends sensitively on the environment. They contribute proportional10% of all galaxies in low density regions of the universe but proportional40% in dense clusters of galaxies. The second major type of galaxy is the "spiral" (or "disc") to which the Milky Way belongs. Spirals have a prominent disc, made of Population I stars and contain a significant amount of gas and dust. The name originates from the distinct spiralarms which exist in many of these galaxies. In low density regions of the universe, proportional80% of the galaxies are spirals while only proportional10% of galaxies in dense clusters are spirals. This is complementary to the behavior of ellipticals. The stars in a disc galaxy are supported against gravity by their systematic rotation velocity. The rotational speed of stars, v(R), at radius Rhas the remarkable property that it remains constant for large R in almost all spirals; the constant value is typically 200 - 300 km s-1. Most discs also contain a spheroidal component of Population II stars. The luminosity of the spheroidal component relative to the disc correlates well with several properties of these galaxies. This fact has been used for classifying the disc galaxies into finer divisions. The stars in a galaxy are not distributed in completely uniform manner. A typical galaxy contains several smaller stellar systems, each containing proportional102 - 106 stars. These systems, usually called star clusters, can be broadly divided into two types called "open clusters" and "globular clusters". Open clusters consist of proportional102 - 103 population I stars bound within a radius of proportional1 - 10 pc. Most of the stars in these clusters are quite young. In contrast, globular clusters are Population II systems with proportional104 - 106 stars. The Milky Way contains proportional200 globular clusters which are distributed in a spherically symmetric manner about the center of the galaxy. Unlike open clusters, the stars in globular clusters are quite old. The number density of stars in the core of a globular cluster, proportional104MSun pc-3, is much higher than that of a typical galaxy, proportional0.05MSun pc-3. The core radius of the globular clusters is proportional1.5 pc while the "tidal radius", which is the radius at which the density drops nearly to zero, is proportional50 pc. In addition to the stars, the Milky Way also contains gas and dust which contribute proportional(5 - 10)% of its mass. This interstellar medium may be roughly divided into a very dense, cold, molecular component (with about 104 particles per cm-3 and a temperature of proportional100 K), made of interstellar clouds, a second component which is atomic but neutral, (with n round 1 cm-3 and T round 103 K), and a third component which is ionized and very hot (with n round 10-3 cm-3 and T round 106 K). Though the interstellar medium is principally made of hydrogen, it also contains numerous other chemical species and an appreciable quantity of tiny solid particles ("dust"). The spiral arms are concentrations of stars and interstellar gas and are characterized by the presence of ionized hydrogen. This is also the region in which young stars are being formed. For a more detailed study of galaxies, one can divide ellipticals and spirals into subsets and also add two more classes of galaxies, called "lenticulars" and "irregulars". The ellipticals are subdivided as (E1, ..., En, ...) where n = 10(a - b)/a with a and b denoting the major and minor axis of the ellipticals. The "lenticulars" (also called SO) are the galaxies "in between" ellipticals and spirals. They have a prominent disc which contains no gas, dust, bright young stars or spiral arms. Though they are smooth and featureless like ellipticals, their surface brightness follows the exponential law of the spirals. They are rare in low density regions (less than 10%) but constitute nearly half of the galaxies in the high density regions. The spirals are subdivided into Sa, Sb, Sc, Sd with the relative luminosity of the spheroidal component decreasing along the sequence. The amount of gas increases and the spiral arm becomes more loosely spiraled along the sequence from Sa to Sd. The Milky Way is between the types Sb and Sc. There also exists another class of spirals called "barred spirals" which exhibit a bar-like structure in the center. They are classified as SBa, SBb etc. Finally, irregulars are the galaxies which do not fall in the above mentioned morphological classification. These are low luminosity, gas rich systems with massive young stars and large HII regions (i.e. regions containing ionized hydrogen). More than one third of the galaxies in the Milky Way's neighborhood are irregulars. They are intrinsically more difficult to detect at larger distances because of their low luminosity. As seen earlier stars evolve and their luminosity and hence the color changes. Since galaxies are made of stars, galaxies will also exhibit color evolution. Besides, the gas content and elemental abundances of the galaxies will change as the stars are formed and end as planetary nebulae or supernovae.

5.5 Quasars

    Most galaxies which are observed have a fairly low redshift (z < 0 - 5) and an extended appearance on a photographic plate. There exists another important class of objects, called "quasars", which exhibit large redshifts (up to z round 5) and appear as point sources in the photographic plate (Shaver et al., 1996). Estimating the distance from the redshift and using the observed luminosity, it is found that quasars must have a luminosity of about Lq round 1046 - 1047 erg s-1. It is possible to estimate the size of the region emitting the radiation from the timescale in which the radiation pattern is changing. It turns out that the energy from the quasar is emitted from a very compact region. One can easily show that nuclear fusion cannot be a viable energy source for quasars. It is generally believed that quasars are fueled by the accretion of matter into a supermassive black hole (M round 108MSun) in the center of the host galaxy. The friction in the accretion disc causes the matter to lose angular momentum and hence spiral into the black hole; the friction also heats up the disc. Several physical processes transform this heat energy into radiation of different wavelengths. Part of this energy can also come out in the form of long, powerful, jets. The innermost regions of the quasar emit x and gamma rays. Outer shells emit UV, optical, and radio continuum radiation in the order of increasing radius. Very bright quasars have an apparent magnitude of mB round 14 (which corresponds to a flux of 10-25 erg s-1 cm-2 Hz-1) while the faintest ones have mB round 23. The absolute magnitudes of the quasars are typically -30 < MB < -23; in contrast, galaxies fall in the band -23 < MB < -16. The relation between m and M for any given quasar depends on the estimated distance to the quasar, which in turn depends on the cosmological model and the quasar's redshift. The luminosity function of the quasars, as a function of their redshift, has been a subject of extensive study. These investigations show that:

  1. the space density of bright galaxies (about 10-2 Mpc-3) is much higher than that of quasars (about 10-5 Mpc-3) with z < 2, and
  2. bright quasars were more common in the past (z round 2) than today (z round 0).

    Quasars serve as an important probe of the high redshift universe. Quasars are believed to be one extreme example of a wide class of objects called "active galaxies". This term denotes a galaxy which seems to have a very energetic central source of energy. This source is most likely to be a black hole powered by accretion. One kind of active galaxy which has been studied extensively are radio galaxies. The most interesting feature about these radio galaxies is that the radio emission does not arise from the galaxy itself but from two jets of matter extending from the galaxy in opposite directions. It is generally believed that this emission is caused by the synchrotron radiation of relativistic electrons moving in the jets. The moving electrons generate two elongated clouds containing magnetic fields which, in turn, trap the electrons and lead to the synchrotron radiation.

 

  5.6 Stellar Evolution

    The time evolution of a star, which can be depicted as a path in the Hertzsprung-Russell diagram, is quite complicated because of many physical processes which need to be taken into account. Detailed calculations, based on the numerical integration of the relevant differential equations, have provided a fairly comprehensive picture of stellar evolution. One of the primary sources of stellar energy is a series of nuclear reactions converting four protons into a helium nucleus. Since the simultaneous collision of four particles is extremely improbable, this process of converting hydrogen into helium proceeds through two different sequences of intermediate reactions, one called proton-proton chain and the other called carbon-nitrogen-oxygen cycle. In the p-p chain, helium is formed through deuterium and 3He in the intermediate steps; this reaction is the dominant mechanism for hydrogen-helium conversion at temperatures below proportional2 cross 107 K. In the CNO cycle, hydrogen is converted into helium through a sequence of steps involving 12C as a catalyst, i.e., the amount of 12C remains constant at the end of the cycle of reactions. Since the Coulomb barrier for carbon nuclei is quite high, the CNO cycle is dominant only at higher temperature. The evolution of a star like the Sun during the phase of hydrogen-helium conversion, called the "main sequence" phase, is fairly stable. The stability is essentially due to the following regulatory mechanism: Suppose the temperature decreases slightly causing the nuclear reaction rate to decrease. This will make gravity slightly more dominant, causing a contraction. Once the star contracts, the temperature will again increase, thereby increasing the rate of nuclear reactions and the pressure support. This will restore the balance. After the burning of core hydrogen ends, the core will undergo a contraction, increasing its temperature; if the star now heats up beyond the helium ignition temperature, then the burning of helium will start and stabilize the star. In principle, such a process can continue with the building up of heavier and heavier elements. But to synthesize elements heavier than He is not straight forward because He has a very high binding energy per nucleon among the light elements. Stars achieve synthesis of post-helium elements through a process known as "triple-alpha reaction" which proceeds as 4He(4He, 8Be)gamma; 8Be(4He, 12C)gamma. Once 12C has been synthesized, production of heavier elements like 16O, 20Ne, 24Mg, etc., can occur through various channels, provided temperatures are high enough, and the star can evolve through successive stages of nuclear burning. The ashes of one stage can become the fuel for the next stage as long as each ignition temperature is reached. Such stars will evolve into structures which contain concentric shells of elements. For example, a 15MSun star, during its final phase can have layers of iron, silicon, oxygen, neon, carbon, helium, and hydrogen all burning at their inner edges.

    The details of the above process, which occurs after the exhaustion of most of the fuel in the core, depend sensitively on the core mass. Consider, for example, a star with M > MSun. Its evolution proceeds in the following manner: Once the hydrogen is exhausted in the core, the core, containing predominantly helium, undergoes gravitational contraction. This increases the temperature of the material just beyond the core and causes renewed burning of hydrogen in a shell-like region. Soon, the core contracts rapidly, increasing the energy production and the pressure in the shell, thereby causing the outer envelope to expand. Such an expansion leads to the cooling of the surface of the star. About this time, convection becomes the dominant mechanism for energy transport in the envelope and the luminosity of the star increases due to convective mixing. This is usually called the "red-giant" phase. During the core contraction, the matter gets compressed to very high densities ( proportional105 g cm-3) so that it behaves like a degenerate gas and not as an perfect gas. Once the core temperature is high enough to initiate the triple-alpha reaction, helium burning occurs at the core. Since degenerate gas has a high thermal conductivity, this process occurs very rapidly, called "helium flash". If the core was dominated by gas pressure, such an explosive ignition would have increased the pressure and led to an expansion; but since the degeneracy pressure is reasonably independent of temperature, this does not happen. Instead the evolution proceeds as a run-away process: The increase in temperature causes an increase in the triple-alpha reaction rate, causing further increase in temperature etc. Finally, when the temperature becomes proportional3.5 cross 108 K, the electrons become non-degenerate; the core then expands and cools. The star has now reached a stage with helium burning in the core and hydrogen burning in a shell around the core. Soon the core is mostly converted into carbon and the reaction again stops. The process described above occurs once again, this time with a carbon-rich, degenerate core, and a helium burning shell. This situation, however, turns out to be unstable because the triple-alpha reaction is highly sensitive to temperature. This reaction can over respond to any fluctuations in pressure or temperature thereby causing pulsations of the star with increasing amplitude. Such violent pulsation can eject the cool, outer layers of the star leaving behind a hot core. The ejected envelope becomes what is known as a "planetary nebula". The above discussion assumes that the core could contract sufficiently to reach the ignition temperature for carbon burning. In low mass stars, degeneracy pressure stops the star from reaching this phase and it ends up as a "white dwarf" supported by the degeneracy pressure of electrons against gravity. A more complicated sequence of nuclear burning is possible in stars with M much greater MSun. After the exhaustion of carbon burning in the core, one can have successive phases with neon, oxygen and silicon burning in the core with successive shells of lighter elements around it. This process can proceed until 56Fe is produced in the core.

    The binding energy per nucleon is maximum for the 56Fe nucleus; hence it will not be energetically feasible for heavier elements to be synthesized by nuclear fusion. The core now collapses violently reaching very high temperatures, about 1010 K. The 56Fe photo-disintegrates into alpha particles, and then even the alpha particles disintegrate at such high temperatures to become protons. The collapse of the core squeezes together protons and electrons to form neutrons and the material reaches near-nuclear densities forming a "neutron star". There exist several physical processes which can transfer the gravitational energy from core collapse to the envelope, thereby leading to the forceful ejection of the outer envelope causing a "supernova explosion". A remnant with smaller mass is left behind. Numerical studies show that stars with M > 8MSun burn hydrogen, helium, and carbon and evolve rather smoothly. During the final phase, such a star explodes as a supernova leaving behind a remnant which could be a white dwarf, neutron star, or black hole. It is generally believed that stars with masses in the intermediate range, 2MSun < M < 8MSun, do not burn hydrogen and helium in degenerate cores but evolve through carbon burning in degenerate matter (for M < 4MSun) ending again in a supernova explosion. Stars with lower mass do not explode but end up as planetary nebulae. Low mass stars with M < 2MSun ignite helium in degenerate cores at the tip of the red giant branch and then evolve in a complicated manner. The synthesis of elements as described above proceeds smoothly up to 56Fe. Heavier elements are formed by nuclei absorbing free neutrons, produced in earlier reactions by two different processes called the "r-process" (rapid neutron capture process) and the "s-process" (slow neutron capture process). During the supernova explosion, a significant part of the heavy elements synthesized in the star will be ejected out into the interstellar space. A second generation of stars can form from these gaseous remnants. The initial composition of material in this second generation will contain a higher proportion of heavier elements (called "metals") compared to the first generation stars. Both these types of stars are observed in the universe; because of historical reasons, stars in the second generation are called population I stars while those in the first generation are called population II stars.

    The above discussion shows how stars could synthesize heavier elements, even if they originally start as gaseous spheres of hydrogen. The study of the spectra of stars allows to determine the relative proportion of various elements present in the stars. Such studies show that population II stars are made of about 75% hydrogen and 25% helium; even population I stars consist of an almost similar proportions of hydrogen and helium with a small percentage of heavier elements (proportional2%). It is possible to show, using stellar evolution calculations, that it is difficult for the stars to have synthesized elements in such a proportion, if they originally had only hydrogen (called primordial or population III stars). Hence, such a universal composition leads us to conjecture that the primordial gas from which population II stars have formed must have been a mixture of hydrogen and helium in the ratio 3:1 by weight. Heavier elements synthesized by these population II stars would have been dispersed in the interstellar medium by supernova explosions. The population I stars are supposed to have been formed from this medium, containing a trace of heavier elements. The helium present in the primordial gas should have been synthesized at a still earlier epoch, the Big Bang, and cannot be accounted for by stellar evolution. Cosmological models provide an explanation for the presence of this primordial helium.