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 m_p and the Bohr radius a₀) of structures down to the planets can be expressed in terms of the electromagnetic fine structure constant _e , Eq.(4), the gravitational fine structure constant _g , Eq.(3), and the electron-to-proton mass ratio (m_p 1837m_e).

    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 R_B when they become bound. After binding, motions in the protogalaxy randomize and equilibrate in the gravitational field of the galaxy at a radius R_B/2. Thereafter, they will deflate on a cooling timescale, with a virial temperature

.

(14)

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

.

(16)

    The free-fall timescale is

,

(17)

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

,

(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 kT_virial > _e²m_ec² at the "magic radius" R_g; that is,

.

(19)

    Gas clouds of mass < M_g cool more efficiently owing to recombination, and can never be pressure supported. Thus, M_g is a characteristic maximum galactic mass. Primordial clouds of mass < M_g 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 M_g above which any fluctuations are likely to remain amorphous and gaseous, and which may thus relate to the mass-scale of galaxies. The quantities M_g and R_g 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.

    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 h/d, where d is their average separation. Provided the electrons are non-relativistic, the degeneracy energy is p²/2m_e, so the virial theorem implies (Dyson, 1972; Weisskopf, 1975; Haubold and Mathai, 1994)

.

(20)

    Here N is the number of protons in the star, N_{0 g}^-3/2, and R N^1/3d 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

,

(21)

    and then decrease, reaching zero when d is

,

(22)

    where r_e is the size of an electron, r_e /m_ec 10^-10 cm; accordingly the electron timescale is defined as t_e = r_e/c 10^-20s. A collapsing cloud becomes a star only if T_max is high enough for nuclear reactions to occur, that is kT_max > qm_ec² where q depends on the strong and electromagnetic interaction constants and is 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 Nm_p and radius R, the ratio of radiation pressure to matter pressure can be shown to be

,

(23)

    so the upper limit to the mass of a star is also N₀m_p. 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,

,

(24)

    where is the opacity. The appropriate 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.

    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 M^-1/3 indicated by Equation (22). However, when M gets so large that kT_max m_ec², the electrons will end up relativistic, with the degeneracy energy being pc rather than p²/2m_e. 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

,

(25)

    which characterises stars in general. A more precise expression for this critical value of M (Chandrasekhar mass) is 5.6²M, where is the number of free electrons per nucleon (Chandrasekhar, 1935). For a star which has burned all the way up to iron, 1/2, and the limiting Chandrasekhar mass, taking into account the onset of inverse ß-decay when electrons get relativistic, is 1.25 M. Stars bigger than M_c 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 m_e m_p, the particles which dominate the neutron star's mass are relativistic. The maximum mass is still of the order of M, 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 m_p/r_p³. 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 nearest large galaxy to the Milky Way is the "Andromeda galaxy" which is about 670 kpc away. Its mass is 3 10¹¹M (or perhaps as much as 1.5 10¹¹M if it has a massive halo) and it has a size of 50 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 2 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 R^1/4 law with an effective radius R_e (1 - 2)h^-1 Mpc (0.5 h 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 17 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 10⁸ 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 3 Mpc while its central core is 600 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 80 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 37 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).

    Galaxies range widely in their sizes, shapes, and masses; nevertheless, one may talk of a typical galaxy as something made out of 10¹¹ stars. Taking the mass of a star to be that of the Sun, the luminous mass in a galaxy is 10¹¹M 2 10⁴⁴ g.

    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.

    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 2 10⁷ 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); ⁸Be(⁴He, ¹²C). 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 15M 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 > M. 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 ( 10⁵ 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 3.5 10⁸ 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 M. 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 > 8M 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, 2M < M < 8M, do not burn hydrogen and helium in degenerate cores but evolve through carbon burning in degenerate matter (for M < 4M) ending again in a supernova explosion. Stars with lower mass do not explode but end up as planetary nebulae. Low mass stars with M < 2M 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 (2%). 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.