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We have summarized some properties of sunspots in [§4.8.](#page-138-1) Now we shall discuss how these properties can be explained with the basic principles of MHD.
First of all, a sunspot is a region of concentrated magnetic field (of order 0.3 T) with very little magnetic field in the surrounding region. Why does th... | {
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"Header 2": "**8.6 Sunspots and magnetic buoyancy**",
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If there is any magnetic field in the solar interior, we saw in [§8.6](#page-253-1) that one can combine the ideas of flux freezing, magnetoconvection and magnetic buoyancy to explain the bipolar sunspots. But why should there be any magnetic field to begin with? Most stars are believed to have magnetic fields. We poin... | {
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"Header 2": "**8.7 A qualitative introduction to dynamo theory**",
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The interstellar medium inside a galaxy is usually found to be distributed rather non-uniformly. [Figure 8.7](#page-259-0) shows how the interstellar medium is distributed in the galaxy M81. In parts of the spiral arms, the interstellar medium seems to form a succession of clumps like beads on a string. It was [Parker]... | {
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"Header 2": "**8.8 Parker instability**",
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We have pointed out in [§8.5](#page-249-1) that the magnetic Reynolds number is very high in most astrophysical systems and the diffusion term η∇2**B** in [\(8.27\)](#page-248-1) can be neglected, leading to the flux freezing condition. It would appear that the diffusion of magnetic fields would be a very unimportant a... | {
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"Header 2": "**8.9 Magnetic reconnection**",
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In the previous few sections, we have considered several important astrophysical applications of MHD, treating the plasma as a continuum. There are some astrophysical plasma problems which require a more microscopic point of view and we have to go beyond MHD. One such problem is to understand why many astrophysical sys... | {
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"Header 2": "**8.10 Particle acceleration in astrophysics**",
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\tag{8.48}$$
This is exactly like the kinetic theory result of finding the probability that the time between two collisions for a particle is in the range t to t+dt and is discussed in any elementary textbook presenting kinetic theory (see, for example, Reif, 1965, §12.1; Saha and Srivastava, 1965, §3.30). Now a part... | {
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"Header 2": "**8.10 Particle acceleration in astrophysics**",
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A very famous result of classical electrodynamics is that accelerated charged particles emit electromagnetic radiation. Whenever the velocity of a charged particle in a plasma changes, we, therefore, expect radiation to come out. In this section and the next, we shall discuss two astrophysically important plasma radiat... | {
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"Header 2": "**8.11 Relativistic beaming and synchrotron radiation**",
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\tag{8.60}$$
The frequency of gyration of an electron of charge e and rest mass $m_e$ gyrating in a magnetic field B is $Be/\gamma m_e$ (see, for example, Jackson, 2001, §12.2), which can be written as $\omega_{g,nr}/\gamma$ , where
$$\omega_{\rm g,nr} = \frac{Be}{m_e} \tag{8.61}$$
is the non-relativistic gy... | {
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"Header 2": "**8.11 Relativistic beaming and synchrotron radiation**",
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Synchrotron radiation discussed in the previous section is an example of *nonthermal radiation*, i.e. a type of radiation arising from a cause other than temperature. The radiation emitted by a body just because of its heat is called *thermal radiation*. In [§2.2](#page-42-4) we have discussed the emission of radiation... | {
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"Header 2": "**8.12 Bremsstrahlung**",
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We end this chapter by pointing out how electromagnetic waves are affected by the presence of plasma. The electric field of the wave accelerates the electrons in the plasma, which then has an effect on the propagation. Because of the inertia of the electrons, very high frequency waves cannot move the electrons much. So... | {
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"Header 2": "**8.13 Electromagnetic oscillations in cold plasmas**",
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If an electromagnetic wave of frequency ω is sent towards a volume of plasma with a plasma frequency ω<sup>p</sup> greater than ω (if ω<ωp), then the electromagnetic wave is not able to pass through this plasma and the only possibility is that it is reflected back.
The plasma frequency of the Earth's ionosphere is ab... | {
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"Header 2": "**8.13 Electromagnetic oscillations in cold plasmas**",
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We have pointed out in [§6.1](#page-172-3) that astronomers in the early twentieth century thought that our Milky Way Galaxy is the entire Universe! Even a small telescope shows many nebulous objects in the sky. The great German philosopher Kant already conjectured in the eighteenth century that some of these nebulae c... | {
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"Header 2": "**9.1 Introduction**",
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Light coming from a typical simple galaxy seems like a composite of light emitted by a large number of stars. A galaxy of this kind is called a *normal galaxy*. We shall discuss the characteristics of such galaxies in this section. Galaxies with more complex properties will be taken up in [§9.4.](#page-295-1)
![](_pa... | {
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[Figure 9.5](#page-285-0) shows velocity dispersions of several elliptical galaxies plotted against their luminosities. The observational data show a reasonably tight correlation corresponding to the Faber–Jackson relation, without too much scatter.
In addition to the differences in appearance and morphology, spiral ... | {
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Does this dark matter lie in the disk beyond the disk of neutral hydrogen or does it form a halo around the galaxy? We do not know the full answer (see the discussion of gravitational lensing in [§13.3.2\)](#page-418-1).
The asymptotic circular speed v<sup>c</sup> in the flat portion of the rotation curve would certa... | {
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"Header 2": "**9.2 Normal galaxies**",
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Suppose a source emitting sound is moving away from you at speed v and the speed of sound through air is *c*s. It is easy to show that the frequency νobs of the sound wave measured by you as an observer will be related to the frequency νem at which the sound is emitted by
$$\frac{v_{\text{obs}}}{v_{\text{em}}} = \fra... | {
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"Header 2": "**9.3 Expansion of the Universe**",
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As we already pointed out, [Hubble](#page-476-26) [\(1922](#page-476-26)) determined the distances of some nearby galaxies by studying Cepheid variables in them, which were taken as standard candles. For galaxies still further away for which Cepheid variables are not discernable, one can take the brightest stars as sta... | {
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A normal galaxy is made up of stars and interstellar matter. Sometimes, however, it is found that a galaxy may additionally have a compact nucleus at its centre giving out copious amounts of radiation in several bands of electromagnetic spectrum from the radio to X-rays. Such a galaxy is called an *active galaxy* and i... | {
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"Header 2": "**9.4 Active galaxies**",
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Since some authors use the terms 'quasar' and 'QSO' almost interchangeably, we shall use the terms radio-loud and radio-quiet quasars to denote quasars which do and do not emit in the radio. Radio-quiet quasars seem much more numerous than radio-loud quasars. Only a few percent of all quasars seem to be radio-emitters.... | {
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"Header 2": "**9.4 Active galaxies**",
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Using [\(5.38\)](#page-167-2) and [\(9.22\),](#page-302-0) we get
$$\frac{c GMm_{\rm H}}{\sigma_{\rm T}} = \left(\frac{2GM}{c^2}\right)^2 \sigma T^4,$$
from which
$$T \approx 3.7 \times 10^5 M_8^{-1/4}. (9.23)$$
It thus follows that more massive black holes are associated with smaller temperatures, which may se... | {
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If the radio jet of an active galaxy has to be within a certain angle with respect to the line of sight in order for the galaxy to appear as a radio-loud quasar and if it appears as a radio galaxy otherwise, then one can think of carrying out similar statistical tests on the populations of radio-loud quasars and radio ... | {
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Like gregarious animals, galaxies also seem to like living in herds. Our Galaxy is a member of a cluster of about 35 galaxies called the *Local Group*. The Andromeda Galaxy M31 is the most prominent member of the Local Group and our Galaxy is the second most prominent member. The members of the Local Group are spatiall... | {
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"Header 2": "**9.5 Clusters of galaxies**",
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On substituting these values in [\(8.71\)](#page-273-7) and assuming that the total volume of the gas is of order 1 Mpc3, it is easy to show that the total X-ray luminosity from the cluster would be 1037 W. This is indeed the typical X-ray luminosity of galaxy clusters. The total mass
... | {
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"Header 2": "**9.5 Clusters of galaxies**",
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Are the galaxy clusters the largest structures in the Universe or are there even bigger structures? To answer this question, it is necessary to map the three-dimensional distribution of many galaxies. We see galaxies distributed over the two-dimensional celestial sphere. Once we find the redshift of a galaxy, we can us... | {
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"Header 2": "**9.6 Large-scale distribution of galaxies**",
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We end our discussion of extragalactic astronomy with a few words about one of the most intriguing and ill-understood objects in astronomy – *gamma ray bursts*, abbreviated as GRBs. These are bursts of γ -rays typically lasting for a few seconds. As of now, there is not yet a consensus amongst astrophysicists how these... | {
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"Header 2": "**9.7 Gamma ray bursts**",
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How the Universe began is a fundamental question which has occupied the minds of men from prehistoric times. However, only after Einstein's formulation of general relativity [\(Einstein](#page-474-31), [1916](#page-474-31)), did it become possible to build theoretical models for the evolution of the Universe. Very simp... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**10.1 Introduction**",
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General relativity provides a field theory of gravity. To explain what is meant by a field theory, let us consider the example of the other great classical field theory with which the reader must be familiar – the theory of electromagnetic fields. We consider two charges *q*<sup>1</sup> and *q*<sup>2</sup> with **r**<s... | {
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"Header 2": "**10.2 What is general relativity?**",
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The intrinsic view of curvature usually conforms to our everyday notion of curvature – but not always! For example, we normally think of the surface of a cylinder as a curved surface. But this surface can be unrolled to a plane and has the same geometric properties as a plane surface. All triangles drawn on a cylindr... | {
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"Header 2": "**10.2 What is general relativity?**",
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After the brief qualitative introduction to general relativity in the previous section, we now discuss the possible structure of spacetime in the Universe. In [§9.6,](#page-312-1) we have introduced the *cosmological principle*, which states that space is homogeneous and isotropic. Now, it is possible for space to be h... | {
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"Header 2": "**10.3 The metric of the Universe**",
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Sometimes it is useful to write the Robertson–Walker metric in terms of the variable $\chi$ used in (10.12), (10.14) and (10.15) rather than r. This is easily seen to be
$$ds^{2} = -c^{2}dt^{2} + a(t)^{2} \left[ d\chi^{2} + S^{2}(\chi)(d\theta^{2} + \sin^{2}\theta \, d\phi^{2}) \right], \tag{10.20}$$
where the fu... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**10.3 The metric of the Universe**",
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We now want to derive an equation describing how the scale factor *a*(*t*) appearing in [\(10.19\)](#page-325-0) evolves with time. We shall use some simple considerations
**Fig. 10.3** Sketch of a spherical shell in a region of spherical expansion.
of Newtonian mechanics. As we alre... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**10.4 Friedmann equation for the scale factor**",
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The three possible values of k basically force a, $\dot{a}$ and $\rho$ to satisfy three possible relationships amongst themselves which follow from (10.27) on substituting the values of k. Since it is the mass-energy which creates the curvature of spacetime in general relativity, we do expect such relationships. On ... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**10.4 Friedmann equation for the scale factor**",
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As pointed out in [§10.4,](#page-327-4) we need to specify how the density ρ of the Universe varies with the scale factor *a* in order to solve for *a* as a function of time. We have to consider the contents of the Universe for this purpose. Before we get into the discussion of the specific contents, let us consider a ... | {
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"Header 2": "**10.5 Contents of the Universe. The cosmic blackbody radiation**",
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[Figure 10.4](#page-334-0) shows the spectrum of CMBR measured by COBE, as reported by [Mather](#page-477-30) *et al.*
**Fig. 10.4** The spectrum of cosmic microwave background radiation (CMBR) as obtained by COBE. From [Mather](#page-477-30) *et al.* [\(1990](#page-477-30)). (c America... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**10.5 Contents of the Universe. The cosmic blackbody radiation**",
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When k = +1, the quadrature (10.55) can be worked out to give
$$a = \frac{4\pi G}{3c^2} \rho_{\text{M},0} a_0^3 (1 - \cos \eta).$$
This can be written as
$$\frac{a}{a_0} = \frac{1}{2} \left( \frac{8\pi G \rho_{\text{M},0}}{3H_0^2} \right) \frac{a_0^2 H_0^2}{c^2} (1 - \cos \eta).$$
On making use of (10.28), (10.... | {
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"Header 2": "**10.5 Contents of the Universe. The cosmic blackbody radiation**",
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We see in [Figure 10.5](#page-338-3) that the solutions for all the three values of *k* behave very similarly at sufficiently early times. We can simplify the solutions for *k* = ±1 when η 1. Both [\(10.56\)](#page-337-3) and [\(10.58\)](#page-338-4) reduce to
$$\frac{a}{a_0} \approx \frac{\Omega_{\rm M,0}}{2|1 - \O... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**10.6.3 Approximate solution for early epochs**",
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On substituting $\rho=\rho_{\rm R}$ as given by (10.48), we obtain from (10.32) that
$$a\dot{a} = \sqrt{\frac{8\pi G\rho_{\rm R,0}}{3}}a_0^2,$$
of which the solution is
$$\frac{a}{a_0} = \left(\frac{32\pi G\rho_{R,0}}{3}\right)^{1/4} t^{1/2}.$$
(10.65)
The radiation-dominated Universe expanded with time as $... | {
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"Header 2": "**10.6.3 Approximate solution for early epochs**",
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- **10.1** Since the recession velocity of a galaxy can be determined reasonably accurately from the redshift in its spectrum, an application of Hubble's law to estimate distance makes the estimated distance uncertain as *h*−<sup>1</sup> if the value of the Hubble constant is uncertain. If the average density of the Un... | {
"Header 1": "**The spacetime dynamics of the Universe**",
"Header 2": "**Exercises**",
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The present uniform expansion of the Universe suggests that there was an epoch in the past when the Universe was in a singular state with infinite density. Since most of the known laws of physics become inapplicable to such a singular state, we cannot extrapolate to earlier times before this epoch of singularity. We th... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.1 Setting the time table**",
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Let us consider a reaction in which *A*, *B*, *C*, . . . combine to produce *L*, *M*, *N*,...:
$$A + B + C + \dots \longleftrightarrow L + M + N + \dots, \tag{11.1}$$
which can proceed in both directions. Under normal circumstances, we expect this reaction to reach a *chemical equilibrium* when the concentrations o... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.2 Thermodynamic equilibrium**",
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Similarly from (11.8) we get
$$\rho = \begin{cases} \frac{g}{2c^2} a_{\rm B} T^4 & \text{(boson),} \\ \frac{7}{8} \frac{g}{2c^2} a_{\rm B} T^4 & \text{(fermion).} \end{cases}$$
(11.10)
Here
$$a_{\rm B} = \frac{\pi^2 \kappa_{\rm B}^4}{15\hbar^3 c^3} \tag{11.11}$$
is the Stefan constant. If we take g = 2 for phot... | {
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"Header 2": "**11.2 Thermodynamic equilibrium**",
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The crucial question is whether these reactions would have been able to reach chemical equilibrium, for which the condition is given by [\(11.3\).](#page-347-5) To answer this question, we need to find out the reaction rate , which depends on the coupling constant of the weak interaction and can be calculated from a kn... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.2 Thermodynamic equilibrium**",
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Various thermal processes in the early Universe created photons and that is why we still have a cosmic background of blackbody radiation. Exactly similarly we would expect a background of neutrinos because neutrinos were created in the early Universe by reactions like [\(5.29\)](#page-156-5) and [\(5.30\)](#page-156-4)... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.4 The cosmic neutrino background**",
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As long as neutrinos are relativistic, the number density is given by [\(11.9\).](#page-348-6) It turns out that [\(11.9\)](#page-348-6) can still be used to calculate the number density of neutrinos even if they had become non-relativistic, provided we take *T* to be a quantity which falls off as *a*−<sup>1</sup> in t... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.4 The cosmic neutrino background**",
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In [§11.3](#page-350-4) we considered the nuclear reaction era when the temperature of the Universe was of order MeV (≈1010 K). The next two sections followed the fate of particles which got decoupled during this era. This era roughly lasted from about 10−<sup>1</sup> s to 102 s after the Big Bang, according to [\(10.6... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.6 Some considerations of the very early Universe**",
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After discussing the physics of the nucleosynthesis era and its consequences in [§11.3](#page-350-4)[–§11.5,](#page-355-1) we took a brief digression in [§11.6](#page-358-1) to raise some theoretical issues pertaining to still earlier times. Now we again follow the evolutionary history of the Universe and look at some ... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.7 The formation of atoms and the last scattering surface**",
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While the mathematical theory of Thomson scattering can be developed by treating the photons as making up a classical electromagnetic wave (as discussed in [§2.6.1\)](#page-69-2), the theory of the Compton effect requires a treatment of photons as particles. The Compton effect becomes important when the photon energy i... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.7 The formation of atoms and the last scattering surface**",
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We can get direct information about some material object in the astronomical Universe if it either emits radiation or absorbs radiation passing through it. We can study the distribution of primordial matter at the moment of its decoupling from radiation by analysing the CMBR which was emitted by this matter. After the ... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.8 Evidence for evolution during redshifts** *z* **∼1–6**",
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Reproduced with permission from *Astrophysical Journal*.)
λL<sup>α</sup> to (1 + *z*em)λL<sup>α</sup> in the spectrum of the quasar corresponding to the full run of possible values of *z*abs. The presence or absence of such an absorption trough in the spectrum of a distant quasar would give us an estimate of the amou... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.8 Evidence for evolution during redshifts** *z* **∼1–6**",
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As discussed in [§11.7.1,](#page-361-1) matter was distributed fairly uniformly at the era *z* ≈ 1100 when matter-radiation decoupling took place, the density perturbations at that era being of the order of 10<sup>−</sup>5. On the other hand, the observations discussed in [§11.8](#page-363-1) suggest that the first sta... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.9 Structure formation**",
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[Figure 11.7](#page-371-0) gives a sketch of how the perturbations must have grown. When the Universe was radiation-dominated till *t* = *t*eq, the perturbations in the baryonic matter and the cold dark matter must have had similar amplitudes and could not grow. Since perturbations in the baryonic matter remained fro... | {
"Header 1": "**The thermal history of the Universe**",
"Header 2": "**11.9 Structure formation**",
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We have pointed out in [§10.2](#page-317-2) that in general relativity we have to deal with the curvature of spacetime and that tensors provide a natural mathematical language for describing such curvature. We now plan to give an introduction to tensor analysis and then an introduction to general relativity at a techni... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "**12.1 Introduction**",
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\frac{\partial x^\nu}{\partial \overline{x}^d}.$$
Using the fact that
$$\frac{\partial \overline{x}^d}{\partial x^\delta} \frac{\partial x^\nu}{\partial \overline{x}^d} = \delta^\nu_\delta,$$
we easily get
$$\overline{T}_{l..d}^{ab..d} = T_{\lambda..\delta}^{\alpha\beta..\delta} \frac{\partial \overline{x}^a}{\... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "**12.1 Introduction**",
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However, if we divide the two vectorial components of velocity by $\sqrt{g_{rr}} = 1$ and $\sqrt{g_{\theta\theta}} = r$ respectively, then we get the contravariant velocity vector $(\dot{r}, \dot{\theta})$ in polar coordinates. In general, if we divide the *i*-th vectorial component of a vector in an orthogonal c... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "**12.1 Introduction**",
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\tag{12.24}$$
We leave it for the reader to argue that the covariant derivative of the tensor $A_{ik}$ must be given by
$$\frac{DA_{ik}}{Dx^l} = \frac{\partial A_{ik}}{\partial x^l} - \Gamma_{kl}^m A_{im} - \Gamma_{il}^m A_{mk}.$$
(12.25)
We now show that the Christoffel symbol is symmetric in its bottom two in... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "**12.1 Introduction**",
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Let us try to write down Stokes's theorem of ordinary vector analysis in tensorial notation. We consider ordinary three-dimensional Cartesian space. An element of area $d\mathbf{s}$ is a pseudovector from which we can construct a tensor
$$df^{ik} = \begin{pmatrix} 0 & ds_z & -ds_y \\ -ds_z & 0 & ds_x \\ ds_y & -ds_... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "**12.1 Introduction**",
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We also note from (12.43) that the curvature scalar is a constant over the surface of the sphere, which is expected from the fact that this surface is uniform. We leave it as an exercise for the reader to show that the scalar curvature for the metric (12.10) is $-2/a^2$ . The metrics (12.9) and (12.10) give the only t... | {
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"Header 2": "**12.1 Introduction**",
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On dividing the metric (12.8) by $ds^2$ , we obtain
$$\left(\frac{dr}{ds}\right)^2 = 1 - r^2 \left(\frac{d\theta}{ds}\right)^2 = 1 - \frac{l^2}{r^2}$$
so that
$$\frac{dr}{ds} = \pm \sqrt{1 - \frac{l^2}{r^2}}. (12.55)$$
Dividing (12.54) by (12.55), we get
$$\frac{d\theta}{dr} = \pm \frac{l/r^2}{\sqrt{1 - \fra... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "**12.1 Introduction**",
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We have finished developing the mathematical machinery necessary for formulating general relativity. Now we are ready to get into a discussion of the physics of general relativity. Since it is a complex and difficult subject, we shall proceed cautiously. Suppose we consider the motion of a non-relativistic particle in ... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "12.3 Metric for weak gravitational field",
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Although we are here discussing the motion of a non-relativistic particle, it may be mentioned that [\(12.61\)](#page-394-2) is the correct expression for the action of a free particle even when the particle moves relativistically (see [Landau and Lifshitz,](#page-477-35) [1975](#page-477-35), §8).
When a weak gravit... | {
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"Header 2": "12.3 Metric for weak gravitational field",
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After discussing how general relativity can be used to study a weak gravitational field, we are now ready to present the complete formulation of general relativity. As pointed out in §10.2, the central equation of general relativity is Einstein's equation telling us how the curvature of spacetime is related to the dens... | {
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"Header 2": "12.4 Formulation of general relativity",
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$$u^{i} = (u^{0}, 0, 0, 0).$$
(12.86)
It then follows from (12.83) that
$$u^0 u_0 = -1. (12.87)$$
On lowering the index k in (12.84) by the usual procedure (12.14), we have
$$\mathcal{T}_k^i = \rho c^2 u^i u_k + P \left( \delta_k^i + u^i u_k \right).$$
On making use of (12.86) and (12.87), this gives
$$\m... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "12.4 Formulation of general relativity",
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#### **Exercises**
**12.1** Fully work out all the components of the Christoffel symbol, the Riemann tensor and the Ricci tensor as well as the scalar curvature for the following metrics
$$ds^{2} = dr^{2} + r^{2} d\theta^{2},$$
$$ds^{2} = a^{2} (d\theta^{2} + \sin^{2}\theta d\phi^{2}),$$
$$ds^{2} = a^{2} (d\c... | {
"Header 1": "**Elements of tensors and general relativity**",
"Header 2": "12.4 Formulation of general relativity",
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While the formulation of general relativity presented in [§12.4](#page-397-4) may appear somewhat formal, a physical theory ultimately has to make contact with the results of measurements. Before considering applications of general relativity, we need to understand how the results of time and length measurements can be... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.1 Time and length measurements**",
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Suppose a periodic signal is sent from point A to point B. Let a pulse be emitted at A at world time $x_e^0$ and reach B at world time $x_r^0$ , the propagation time being $x_r^0 - x_e^0$ . Suppose the next pulse is emitted at A at world time $x_e^0 + T^0$ . For a constant gravitational field, the propagation time... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.1 Time and length measurements**",
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This also follows from the fact that the metric at such large distances can be approximated by [\(13.12\),](#page-410-3) which leads to the same results as what we would get from the Newtonian theory of gravity (see [§12.3\)](#page-392-1).
We would expect the metric around a black hole to be given by [\(13.13\).](#pa... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.1 Time and length measurements**",
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A particle will move along a geodesic in the Schwarzschild metric. One can use the geodesic [equation \(12.51\)](#page-391-1) to study the motions of particles. We shall, however, present a discussion starting more from the basics.
Since the Schwarzschild metric is spherically symmetric, a particle moving in this met... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.3.1 Particle motion in Schwarzschild geometry. The perihelion precession**",
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The classical Kepler problem also gives rise to a one-dimensional equation exactly similar to [\(13.20\),](#page-412-3) except that the effective potential does not have the last term −*Ml*2/*r* <sup>3</sup> appearing in [\(13.21\)](#page-412-4) (see, for example, [Goldstein](#page-475-17), [1980](#page-475-17), §3–3; ... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.3.1 Particle motion in Schwarzschild geometry. The perihelion precession**",
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Let us now try a solution of the form
$$u = u_0 + u_1, (13.33)$$
where $u_0$ is given by (13.32). On substituting this in (13.30) and approximating the small perturbation term as $3Mu^2 \approx 3Mu_0^2$ , we get
$$\frac{d^2u_1}{d\phi^2} + u_1 = 3Mu_0^2.$$
On substituting for $u_0$ from (13.32), we get
$$... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.3.1 Particle motion in Schwarzschild geometry. The perihelion precession**",
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ag{13.44}$$
**Fig. 13.3** A massless particle approaching the central mass M from a large distance with an impact parameter h.
We now consider a massless particle approaching the central mass M from a large distance with the impact parameter h as shown in Figure 13.3. The x axis is ch... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.3.1 Particle motion in Schwarzschild geometry. The perihelion precession**",
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Many images of extragalactic sources in the forms of extended arcs are known, suggesting that gravitational lensing is a quite common phenomenon in the extragalactic world.
Let us also comment on another kind of gravitational lensing. As we have discussed in [§9.2.2,](#page-284-6) the rotation curves of spiral galaxi... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.3.1 Particle motion in Schwarzschild geometry. The perihelion precession**",
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We pointed out in §10.2 that one of the implications of an action-at-a-distance theory is that the interaction has to propagate at an infinite speed. Our hope is that this problem would be rectified in a field theory. It is indeed a consequence of general relativity that gravitational interaction propagates at speed c.... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "13.4 Linearized theory of gravity",
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If $h_{ik}$ or $\overline{h}_{ik}$ transformed like tensors, then their divergences would transform like vectors, and it will not be possible to make this divergence zero in a frame when it is non-zero in other frames. Our aim now is to choose a coordinate system in which
$$\frac{\partial \overline{h}'_{ik}}{\par... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "13.4 Linearized theory of gravity",
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It is clear from [\(13.74\)](#page-429-5) that a sudden change in the energy-momentum tensor T*ik* would give rise to a signal propagating away at speed *c*. We also note that in a region of empty space [\(13.73\)](#page-429-6) reduces to the wave equation
$$\Box^2 \overline{h}_{lm} = 0$$
suggesting the possibility... | {
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"Header 2": "**13.5 Gravitational waves**",
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It should be clear that differentiation with respect to $x^\beta$ gives zero and we can write
$$R^{\alpha}_{0\beta0} = -\frac{\partial \Gamma^{\alpha}_{0\beta}}{\partial x^0}.$$
(13.87)
The expression for the Christoffel symbol for the weak gravitational field is given by (13.58). Keeping in mind that $h_{0\beta... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.5 Gravitational waves**",
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[Hint: First argue that
$$\tan \alpha = \left(1 - \frac{2M}{r_i}\right)^{1/2} r_i \left(\frac{d\phi}{dr}\right)_i,$$
where $(d\phi/dr)_i$ is the initial value of $d\phi/dr$ along the light path when the light signal starts at $r_i$ . Then you have to relate $\alpha$ to b=l/e by making use of (13.38). Finally... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**13.5 Gravitational waves**",
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#### 14.1 The basic equations
We have seen in Chapter 10 that certain aspects of spacetime dynamics of the Universe can be studied without a detailed technical knowledge of general relativity. However, some important topics in cosmology – especially those dealing with the analysis of high redshift observations – requ... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "Relativistic cosmology",
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It is possible to put an extra term in Einstein's equation, which would make it
$$G_{ik} = \frac{8\pi G}{c^4} \mathcal{T}_{ik} - \frac{\Lambda}{c^2} g_{ik}.$$
(14.9)
It follows from (12.27) that $g_{ik}$ is also a divergenceless tensor like $G_{ik}$ or $T_{ik}$ . So, on taking the divergence of (14.9), each te... | {
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The fact that $\Omega_{\Lambda,0}$ cancels out of the equation justifies our assertion at the end of §10.6.2 that, even if $\Lambda$ is non-zero, we do not make too much error in many calculations involving earlier times if we use the cosmological solution with $\Lambda=0$ . The values of $\Omega_{\rm M,0}$ and ... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "14.2 The cosmological constant and its significance",
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We have already pointed out in [§13.3.2](#page-418-1) that a light signal travels along a null geodesic, i.e. a special geodesic along which *ds*<sup>2</sup> = 0. We now want to consider the propagation of a light signal from a distant galaxy to us. Let us take our position to be the origin of our coordinate system and... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**14.3 Propagation of light in the expanding Universe**",
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Then (14.28) and (14.29) imply
$$\cos \eta_0 = \frac{2 - \Omega_{\text{M},0}}{\Omega_{\text{M},0}}, \ \sin \eta_0 = \frac{2\sqrt{\Omega_{\text{M},0} - 1}}{\Omega_{\text{M},0}}.$$
(14.30)
In the k = +1 case we are considering, we have $r = \sin \chi$ . Then (14.25) implies
$$r = \sin(\eta_0 - \eta) = \sin \eta_0 ... | {
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"Header 2": "**14.3 Propagation of light in the expanding Universe**",
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Substituting for r from (14.32) in (14.36), we get
$$H_0 d_{\mathcal{L}} = \frac{c}{\Omega_{\mathcal{M},0}^2} [2\Omega_{\mathcal{M},0} z + (2\Omega_{\mathcal{M},0} - 4)(\sqrt{\Omega_{\mathcal{M},0} z + 1} - 1)]. \tag{14.37}$$
This is the functional relationship between $d_L$ and z, telling us what would be the ... | {
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Before discussing how we relate r to z when $\Lambda\neq 0$ , we point out that the surface brightness would still fall as $(1+z)^{-4}$ in accordance with (14.42).
When the cosmological constant $\Lambda$ is non-zero, it is not possible to write down an analytical expression relating r with z. The relation betwe... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**14.3 Propagation of light in the expanding Universe**",
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The lower two panels plot the deviation of *m* − *M* from the solid line for <sup>M</sup>,<sup>0</sup> = 0, -,<sup>0</sup> = 0, showing the data of two groups separately: filled squares for the data of [Riess](#page-479-39) *et al.* [\(1998](#page-479-39)) and open squares for the data of [Perlmutter](#page-478-37) *et... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**14.3 Propagation of light in the expanding Universe**",
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It follows from [\(14.56\)](#page-458-0) that a larger value of <sup>M</sup>,<sup>0</sup> would make θ larger, causing the peak in [Figure 14.5](#page-457-0) to shift leftward. The position of the peak would thus give the value of <sup>M</sup>,0.
When we assume - = 0, the analysis becomes much more complicated and ha... | {
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```
= 3.00 \times 10^8 \,\mathrm{m \ s^{-1}}
Speed of light
= 6.67 \times 10^{-11} \, \mathrm{m}^{3} \, \mathrm{kg}^{-1} \, \mathrm{s}^{-2}
Gravitational constant
= 6.63 \times 10^{-34} \,\mathrm{J}\,\mathrm{s}
Planck constant
= 1.38 \times 10^{-23} \,\mathrm{J} \,\mathrm{K}^{-1}
Boltzmann constant
\kappa_{\mathrm{B}}
... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "A. I Physical constants",
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The Nobel Prize was not awarded to astrophysicists in the early decades of the twentieth century. The most impact-making astrophysicists of that period such as Eddington and Hubble did not win Nobel Prizes. From the middle of the twentieth century, several Nobel Prizes have been given for important discoveries in astro... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Astrophysics and the Nobel Prize**",
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The reader of this book is assumed to have a background of physics appropriate for the advanced undergraduate or the beginning graduate level. Knowledge is assumed of all the standard branches of physics which are usually covered at that level – classical mechanics, electromagnetic theory, special relativity, optics, t... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Suggestions for further reading**",
"token_count": 287,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
There are several excellent elementary astronomy textbooks suitable for beginning undergraduate students where the authors assume very little knowledge of physics and not even a knowledge of calculus, since elementary astronomy courses at this level are popular in many undergraduate programmes. The pioneering classic a... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**General references**",
"token_count": 1214,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
Since books on different astrophysical systems will be mentioned in the references for the various following chapters, here we only mention books which discuss how we obtain astronomical data with the help of the appropriate instruments. Perhaps [Kitchin](#page-476-52) [\(2003](#page-476-52)), [Roy and Clarke](#page-47... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Chapter 1**",
"token_count": 208,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
The standard graduate textbook on radiative processes in astrophysics is the superbly written volume by [Rybicki and Lightman](#page-479-8) [\(1979](#page-479-8)), which is justly regarded as a classic. Since this book mainly deals with well-established principles, it has not dated with time. Other books covering this ... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Chapter 2**",
"token_count": 216,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
Since the study of stars has been the central theme in modern astrophysics for several decades, it is no wonder that there are many excellent books on stellar astrophysics. The classic volumes by Eddington (1930) and [Chandrasekhar](#page-473-8) [\(1939](#page-473-8)), which played very important roles in the historica... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Chapters 3–4**",
"token_count": 564,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
The standard graduate textbook on galactic astronomy by [Binney and Merrifield](#page-473-17) [\(1998\)](#page-473-17) is supposed to be a replacement of the earlier volume by [Mihalas and Binney](#page-477-11) [\(1981](#page-477-11)). In order to make room for the discussion of external galaxies, [Binney and Merrifiel... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Chapter 6**",
"token_count": 608,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
The brilliant small volume by [Weinberg](#page-481-16) [\(1977](#page-481-16)) is a masterpiece of popular science and may be read profitably before delving into the more technical tomes. Somehow cosmology has been a popular subject for textbook writers and there is probably no other branch of astrophysics in which so ... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Chapters 10–11**",
"token_count": 364,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
While general relativity also has been a favourite subject for textbook writers, the last few chapters of [Landau and Lifshitz](#page-477-46) [\(1975\)](#page-477-46) still provide one of the most beautiful and elegant introductions to this subject. [Schutz](#page-479-45) [\(1985](#page-479-45)) and [Hartle](#page-475-... | {
"Header 1": "**Some applications of general relativity**",
"Header 2": "**Chapters 12–14**",
"token_count": 327,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/42e171591d83f3afd34f7952be5782b0.pdf"
} |
Stars: Setting the Stage
Everybody knows what stars are: twinkling light spots in the night sky. In reality, they are huge spheres of gas at high temperature that emit a quantity of light that we can hardly imagine. The properties of stars are described by quite simple laws of physics. To understand the stars and the... | {
"Header 1": "Chapter 1",
"token_count": 745,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The chemical composition of the outer layers of the Sun and stars has been determined spectroscopically. This reflects the composition of the interstellar clouds from which the stars are formed. The composition is roughly the same for all stars in the disk of the Milky Way. The abundances of the 10 most abundant elemen... | {
"Header 1": "1.2 The Chemical Composition of the Sun and Stars",
"token_count": 677,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Due to stellar evolution, a star becomes highly structured, both chemically and physically. For discussing the evolution of stars, it is useful to distinguish several regions in stars.
- The core is the central region of a star where fusion occurs or has occurred.
- The shell zone indicates that fusion occurs or has ... | {
"Header 1": "1.2 The Chemical Composition of the Sun and Stars",
"Header 3": "1.3 The Structure of Stars",
"token_count": 1134,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. The Sun is a good example of an average star. It is about a million times larger in volume than the Earth and it has about a million times more mass. It radiates so brightly that, even at a distance of 150 million km, the tiny fraction of its radiation that hits the Earth is enough to sustain life.
- 2. The age of... | {
"Header 1": "1.5 Summary",
"token_count": 260,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
In this section, we briefly review observations of the fundamental parameters of stars. These are mass (M), radius (R), luminosity (L), and effective temperature $(T_{\rm eff})$ . The existence of observed relations between these parameters provides important information about the internal structure of stars. Any reli... | {
"Header 1": "Observations of Stellar Parameters",
"token_count": 1268,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
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