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Designed for teaching astrophysics to physics students at advanced undergraduate or beginning graduate level, this textbook also provides an overview of astrophysics for astrophysics graduate students, before they delve into more specialized volumes.
Assuming background knowledge at the level of a physics major, the ... | {
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Particle physics, condensed matter physics and astrophysics are arguably the three major research frontiers of physics at the present time. It is generally thought that a physics student's training is not complete without an elementary knowledge of particle physics and condensed matter physics. Most physics departments... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**Preface**",
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In discussing astrophysical topics, one often has to combine results from different branches of physics. Historically these branches may have evolved independently and sometimes the same symbol is used for different things in these different branches. In the case of a few symbols, I have added a subscript to make them ... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**A note on symbols**",
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Astrophysics is the science dealing with stars, galaxies and the entire Universe. The aim of this book is to present astrophysics as a serious science based on quantitative measurements and rigorous theoretical reasoning.
The standard units of mass, length and time that we use (cgs or SI units) are appropriate for ou... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.1 Mass, length and time scales in astrophysics**",
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Astrophysicists have to deal with very different time scales. On the one hand, the age of the Universe is of the order of a few billion years. On the other hand, there are pulsars which emit pulses periodically after intervals of fractions of a second. There is no special unit of time. Astrophysicists use years for lar... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "*Unit of time*",
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From the dawn of civilization, human beings have wondered about the starry sky. Astronomy is one of the most ancient sciences. Perhaps mathematics and medicine are the only other sciences which can claim as ancient a tradition as astronomy. But modern astrophysics, which arose out of a union between astronomy and physi... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.2 The emergence of modern astrophysics**",
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This means that the point *P* in [Figure 1.2](#page-26-1) traces out an approximate circle in the celestial sphere slowly in about 25,800 years, around the pole *K* of the ecliptic. This phenomenon is called *precession* and was discovered by Hipparchus (second century BC) by comparing his observations with the observa... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.2 The emergence of modern astrophysics**",
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Suppose we have two series of lamps – the first series with lamps having intensities *I*0, 2*I*0, 3*I*0, 4*I*<sup>0</sup> . . . , whereas the lamps in the second series have intensities *I*0, 2*I*0, 4*I*0, 8*I*<sup>0</sup> . . . . When we look at the two series of lamps, it is the second series which will appear to hav... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.4 Magnitude scale**",
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Astrophysics is a supreme example of applied physics. To be a competent astrophysicist, first and foremost one has to be a competent physicist. Virtually all branches of physics are needed in the study of astrophysics. Classical mechanics, electromagnetic theory, optics, thermodynamics, statistical mechanics, fluid dyn... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.5 Application of physics to astrophysics. Relevance of general relativity**",
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In most branches of science, controlled experiments play a very important role. Astrophysics is a peculiar science in which astronomical observations take the place of controlled experiments. An astronomer can only observe an astronomical object with the help of the signals reaching us from the object. We list below fo... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.6 Sources of astronomical information**",
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We now consider astronomy with electromagnetic radiation, which is so far our main source of astronomical information. The Earth's atmosphere is an annoying inconvenience for the astronomer. The atmosphere is transparent to only small bands of electromagnetic radiation. Even though visible light passes through the atmo... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.7 Astronomy in different bands of electromagnetic radiation**",
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Since radio waves are not affected by the atmospheric turbulence (though radio waves at wavelengths longer than 20 cm are affected by the plasma irregularities in the ionosphere and the solar wind), the resolving power of a radio telescope is not limited by atmospheric seeing and can achieve the theoretical value given... | {
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"Header 2": "**1.7 Astronomy in different bands of electromagnetic radiation**",
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Somebody embarking on a first study of astronomy may get confused by the names of various astronomical objects. Only a few of the brightest stars were given names in various ancient civilizations. Some of these names are still in use. For stars which do not have names and for all other astronomical objects, astronomers... | {
"Header 1": "**Astrophysics for Physicists**",
"Header 2": "**1.8 Astronomical nomenclature**",
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As we pointed out in [§1.6,](#page-31-2) most of our knowledge about the astrophysical Universe is based on the electromagnetic radiation that reaches us from the sky. By analysing this radiation, we infer various characteristics of the astrophysical systems from which the radiation was emitted or through which the rad... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.1 Introduction**",
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#### **2.2.1 Radiation field**
Let us first consider how we can provide the mathematical description of radiation at a given point in space. It is particularly easy to give a mathematical description of blackbody radiation, which is homogeneous and isotropic inside a container. We shall assume the reader to be famili... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.2 Theory of radiative transfer**",
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by *dA*<sup>1</sup> at *dA*2, then according to [\(2.2\)](#page-43-1) the radiation falling on *dA*<sup>2</sup> in time *dt* after passing through *dA*<sup>1</sup> is
$$I_{\nu 2} dA_2 dt d\Omega_2 d\nu.$$
From considerations of symmetry, this should also be equal to
$$I_{\nu 1} d... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.2 Theory of radiative transfer**",
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from $s_0$ to s along the ray path), we get
$$I_{\nu}(\tau_{\nu}) = I_{\nu}(0) e^{-\tau_{\nu}} + \int_{0}^{\tau_{\nu}} e^{-(\tau_{\nu} - \tau_{\nu}')} S_{\nu}(\tau_{\nu}') d\tau_{\nu}'. \tag{2.20}$$
This is the general solution of the radiative transfer equation.
If matter through which the radiation is passing... | {
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"Header 2": "**2.2 Theory of radiative transfer**",
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By assuming thermodynamic equilibrium, we have derived the tremendously important result [\(2.25\)](#page-49-8) that the source function should be equal to the blackbody function *B*<sup>ν</sup> (*T* ). In a realistic situation, we rarely have strict thermodynamic equilibrium. The temperature inside a star is not const... | {
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"Header 2": "**2.3 Thermodynamic equilibrium revisited**",
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Several treatises have been written on radiative transfer theory, one of the most famous being by [Chandrasekhar](#page-473-3) [\(1950\)](#page-473-3). Now, we have written down the general solution of the radiative transfer equation in [\(2.20\).](#page-48-3) If it is so easy to write down the general solution, then w... | {
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$$S_{\nu}(t_{\nu}) = B_{\nu}(\tau_{\nu}) + (t_{\nu} - \tau_{\nu}) \frac{dB_{\nu}}{d\tau_{\nu}} + \dots$$
(2.37)
Truncating (2.37) after the linear term and substituting it in both (2.35) and (2.36), we get for both positive and negative $\mu$ the very important equation
$$I_{\nu}(\tau_{\nu}, \mu) = B_{\nu}(\tau... | {
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Just similar to (2.39), (2.40) and (2.41), we write down the total energy density, total radiation flux and total radiation pressure integrated over all frequencies:
$$U = \frac{2\pi}{c} \int_{-1}^{1} I \, d\mu, \tag{2.53}$$
$$F = 2\pi \int_{-1}^{1} I \,\mu \,d\mu,\tag{2.54}$$
$$P = \frac{2\pi}{c} \int_{-1}^{1} I... | {
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On the other hand, the ray coming from an off-centre point must emerge from the solar surface at an angle $\theta = \cos^{-1} \mu$ with the vertical, as seen in Figure 2.5, and the corresponding specific intensity will be $I(0, \mu)$ .
Hence (2.70) gives the variation of intensity on the solar disk as we move from... | {
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In [§2.4,](#page-54-1) we have discussed radiative transfer in the outer layers of a star. Astrophysicists studying stellar interiors have to consider radiative transfer in the stellar interior as well. In a typical star, energy is usually produced by nuclear reactions in the innermost core of the star. This energy in ... | {
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"Header 2": "**2.5 Radiative energy transport in the stellar interior**",
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To build a model of the stellar interior, it is necessary to solve a slightly modified version of [\(2.78\)](#page-66-5) as discussed in the [§3.2.3.](#page-85-1) To solve this equation, we need to know the value of opacity χ. The gas in the interior of a star exists under such conditions of temperature and pressure wh... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.6 Calculation of opacity**",
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\tag{2.82}$$
This is the celebrated *Rayleigh scattering*, in which the cross-section goes as ω<sup>4</sup> or as λ<sup>−</sup>4, where λ is the wavelength of the incident electromagnetic wave. Rayleigh scattering provides explanations for many natural as well as astronomical phenomena. In the visible spectrum, blue ... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.6 Calculation of opacity**",
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In [§2.4.3](#page-62-1) we have given a qualitative idea of how spectral lines form. Astronomers, however, require a quantitative theory of spectral lines in order to analyse them to determine the composition of the source. A quantitative theory of spectral lines in a stellar atmosphere involves certain difficulties be... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.7 Analysis of spectral lines**",
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We close our discussion on the interaction of radiation with matter by working out a curious example. Suppose the energy generation rate at the centre of the Sun were to increase or decrease suddenly due to some reason. We expect that eventually the surface of the Sun will become brighter or dimmer as a consequence of ... | {
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At the beginning of [§2.4,](#page-54-2) we pointed out the scope of the subject *stellar interior*. It appears from observational data (to be discussed in detail later) that various quantities pertaining to stars have some relations amongst each other. For example, a more massive star usually has a higher luminosity an... | {
"Header 1": "**Stellar astrophysics I: Basic theoretical ideas and observational data**",
"Header 2": "**3.1 Introduction**",
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We now establish the basic equations of stellar structure by assuming the star to be spherically symmetric. If the star is rotating sufficiently rapidly, then there will be some flattening in the direction of the rotation axis. Again, if the star has strong magnetic fields, that can be another cause of departure from s... | {
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"Header 2": "**3.2 Basic equations of stellar structure**",
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\tag{3.7}$$
The right-hand side is clearly the total gravitational energy *E*<sup>G</sup> of the star, i.e.
$$E_{\rm G} = \int_0^R \left( -\frac{GM_r}{r} \right) 4\pi \rho \, r^2 dr. \tag{3.8}$$
Since (3/2)κB*T* is the mean energy of thermal motion per particle in a region of temperature *T* and hence (3/2)*n*κB*... | {
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We shall derive this alternative equation in the next subsection.
It may be noted that the first three equations of stellar structure -(3.1), (3.2) and (3.15) – follow from fairly straightforward considerations. Only (3.16), which was obtained by Eddington (1916), is somewhat non-trivial. It may be useful for the rea... | {
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We have already derived all the necessary equations for constructing stellar models. Let us now see how it can be done.
First of all, one has to specify the chemical composition of a star, since opacity and the nuclear energy generation rate depend on the chemical composition. The chemical composition can be given by... | {
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Although this method works, it is not a particularly efficient method. A more efficient numerical algorithm was developed by [Henyey, Vardya and Bodenheimer](#page-476-9) [\(1965](#page-476-9)) and is known as the *Henyey method*. This is a standard method widely used in solving stellar structures and is described in s... | {
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"Header 2": "**3.3 Constructing stellar models**",
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We shall now do a few drastic things with the stellar structure [equations \(3.25\)–](#page-90-2) [\(3.28\)](#page-90-3) to extract some relations amongst various quantities pertaining to a star. Since some of our steps will be highly questionable in nature, we shall have to take the derived results with a degree of ca... | {
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"Header 2": "**3.4 Some relations amongst stellar quantities**",
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In the previous section, we arrived at some theoretical conclusions about how various quantities connected with stars may be related to each other. Are these conclusions borne out by observational data? Before we can answer this question, we discuss briefly how various stellar parameters are determined.
#### **3.5.1 ... | {
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Plotting luminosities and masses of such stars, we get [Figure 3.4.](#page-100-0) Our simple theoretical considerations led to [\(3.37\),](#page-94-3) implying that luminosity should go as the cube of mass. The fact that a straight line fits the data reasonably well implies that *L* indeed goes as *Mn*, the index *n* b... | {
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Although most of the data points in Figure 3.5 lie on the diagonal strip called the main sequence, there are also many data points in the upper right corner
**Table 3.1** The relationship amongst colour index *B* − *V*, absolute visual magnitude *M*V, effective surface temperature *T*eff and absolute bolometric magni... | {
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The momentum associated with this energy can be obtained by dividing this by c, which will give us the momentum absorbed per unit time in a unit volume, which is nothing but the force exerted on this unit volume. The star will be able to hold on to this outer layer of gas only if the inward force of gravity is stronger... | {
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- **3.1** Estimate the total thermal energy of the Sun from the fact that its internal temperature is of order 107 K. Show that this is of the same order as the rough estimate of gravitational potential energy.
- **3.2** If the Sun was producing its energy by slow contraction as suggested by Helmholtz and Kelvin, estim... | {
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"Header 2": "**Exercises**",
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We have seen in the previous chapter that many aspects of stellar structure can be understood without a detailed knowledge of stellar energy generation mechanisms. This is indeed fortunate because not much was known about energy generation mechanisms when Eddington was carrying out his pioneering investigations of stel... | {
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"Header 2": "**4.1 The possibility of nuclear reactions in stars**",
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Suppose a nucleus having charge *Z*1*e* can react with a nucleus having charge *Z*2*e*, their number densities per unit volume being *n*<sup>1</sup> and *n*2. We want to calculate the rate of the reaction, i.e. the number of reactions taking place per unit volume per unit time.
If both types of nuclei have a Maxwelli... | {
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"Header 2": "**4.2 Calculation of nuclear reaction rates**",
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\tag{4.16}$$
From (4.10) (keeping in mind that $\tau$ goes as $T^{-1/3}$ ), we now have
$$\langle \sigma v \rangle \propto \frac{S(E_0)}{T^{2/3}} \exp \left[ -3 \left( \frac{e^4}{32\epsilon_0^2 \kappa_B \hbar^2} \frac{m Z_1^2 Z_2^2}{T} \right)^{1/3} \right].$$
(4.17)
Once S(E) for the nuclear reaction is found... | {
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"Header 2": "**4.2 Calculation of nuclear reaction rates**",
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This series of reactions, known as the *CNO cycle*, was independently suggested by von Weizsacker (1938) and ¨ [Bethe](#page-473-11) [\(1939\)](#page-473-11). The reactions in this cycle are the following:
$$^{12}C + ^{1}H \longrightarrow ^{13}N + \gamma,$$
$$^{13}N \longrightarrow ^{13}C + e^{+} + \nu,$$
$$^{13}... | {
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We explained in [§3.3](#page-89-1) how detailed stellar models are calculated. One of the important inputs in a stellar model calculation is the nuclear energy generation rate. We have seen in [§4.2](#page-112-3) and [§4.3](#page-116-2) how this rate can be determined. So we now understand in principle how a stellar mo... | {
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"Header 2": "**4.4 Detailed stellar models and experimental confirmation**",
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This subject, which is the study of solar oscillations, began when [Leighton,](#page-477-6) [Noyes and Simon](#page-477-7) [\(1962\)](#page-477-6) discovered that the surface of the Sun is continuously oscillating with periods of the order of a few minutes. We know that an air column in a pipe vibrates only at some eig... | {
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"Header 2": "**4.4.1 Helioseismology**",
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It became of utmost importance to detect the low-energy *pp* neutrinos, since the predicted theoretical flux is independent of the solar model. Low-energy neutrinos induce the following reaction in gallium
$$^{71}$$
Ga + $\nu \to ^{71}$ Ge + $e^-$ . (4.31)
Hence one can use gallium as a detector of *pp* neutrin... | {
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We pointed out in [§4.3](#page-116-1) that a main-sequence star is expected to generate energy steadily as long as hydrogen in the core is converted into helium. The luminosity or the surface temperature of the star does not change much during this phase when it lies on the main sequence. Eventually, the hydrogen in th... | {
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[Figure 4.11](#page-131-1) shows some of the equipotential surfaces in a typical case. The surface of a star should be an equipotential surface, if we want to ensure that there are no unbalanced horizontal forces at the stellar surface. Each of the stars should extend up to some equipotential surface.
We notice in [F... | {
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When a star becomes a red giant, the gravitational attraction at its inflated surface becomes much smaller than that at an ordinary stellar surface. This reduces the star's ability to hold on to the material on its surface and the surface material may keep escaping. Even in the case of an ordinary star like the Sun, ma... | {
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Chinese astronomers recorded that in the year 1054 a star in the Taurus constellation became so bright that it was visible during daytime. [Figure 4.13](#page-135-0) shows what a modern telescope finds in that spot of the sky. We see a luminous gas shell, known as the Crab Nebula because of its crab-like appearance. By... | {
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In our discussion of stellar structure, we have assumed spherical symmetry. There are two factors which could cause departures from spherical symmetry of a star – rotation and magnetic field. We know quite a lot about the rotation and magnetic field of our nearest star – the Sun. Within the last few years, our knowledg... | {
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The study of planetary motions played a key role in the historical development of astronomy. The study of physical characteristics of planets, however, has now become a branch of science quite distinct from astrophysics and is usually referred to as *planetary science*. In this book, we do not get into a discussion of ... | {
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**4.1** Consider a nucleus of charge *Z*1*e* approaching another nucleus of charge *Z*2*e* with the energy of relative motion equal to *E*. According to classical physics, the nuclei should not be able to come closer than a distance *r*<sup>1</sup> given by
$$E = \frac{1}{4\pi\epsilon_0} \frac{Z_1 Z_2 e^2}{r_1}.$$
... | {
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We have seen in the previous two chapters that the gravitational attraction inside a normal star is balanced by the thermal pressure caused by the thermonuclear reactions taking place in the stellar interior. Eventually, however, the nuclear fuel of the star is exhausted and there is no further source of thermal pressu... | {
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The pressure in a gas arises from the random motions of the particles constituting the gas. If 4π *f* (*p*)*p*2*dp* is the number of particles having momentum between *p* and *p* + *dp* (assuming the distribution function to be isotropic), whereas v is the velocity of a particle having momentum *p*, then the pressure *... | {
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When the electrons are fully relativistic, we can write
$$\sqrt{p^2c^2 + m_{\rm e}^2c^4} \approx pc$$
so that (5.5) gives
$$P = \frac{2\pi c}{3h^3} p_{\rm F}^4.$$
On substituting from (5.8), we have
$$P = K_2 \rho^{4/3}, \tag{5.11}$$
where $K_2$ is given by
$$K_2 = \frac{3^{1/3}}{8\pi^{1/3}} \frac{hc}... | {
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It should be clear from the previous section that the equation of state of degenerate matter relates pressure with density (i.e. it does not involve temperature). Suppose we now want to calculate the structure of a star entirely made of degenerate matter (such as a white dwarf). The equations [\(3.25\)](#page-90-2) and... | {
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This figure is adapted from Chandrasekhar (1984), where the unit of radius $l_1$ used on the vertical axis is defined.
which combine to give
$$R \propto M^{-1/3}.\tag{5.25}$$
This is the very important mass–radius relation of white dwarfs within which matter satisfies the non-relativistic equation of state (5.9... | {
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Just as the degeneracy pressure of electrons supports a white dwarf against gravity, the degeneracy pressure of neutrons supports a neutron star. According to astrophysical folklore, on hearing of the discovery of the neutron in Cavendish Laboratory (Chadwick, 1932), Landau immediately suggested that there can be stars... | {
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A definitive observational confirmation for the existence of neutron stars came when [Hewish](#page-476-22) *et al.* [\(1968\)](#page-476-22) discovered radio sources which were giving out radio pulses at intervals of typically a second. The signal from such a source
**Fig. 5.3** Radio ... | {
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However, even on purely dimensional grounds, we expect something like (5.35) to hold at least approximately.
#### 5.5.1 The binary pulsar and testing general relativity
We now discuss a very intriguing object which was first discovered by Hulse and Taylor (1975). They found a pulsar with a mean period of 0.059 s. H... | {
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[Backer](#page-472-10) *et al.* [\(1982\)](#page-472-10) discovered a pulsar with a period of 1.56 ms, which was considerably shorter than the period of any pulsar known at that time. The pulsar with the second shortest period known at that time, the Crab pulsar, had a period of 33.1 ms. Subsequently several other puls... | {
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A second kind of evidence for the existence of neutron stars started coming at about the same time when pulsars were discovered. [Giacconi](#page-475-2) *et al.* [\(1962](#page-475-2)) discovered several celestial X-ray sources with the help of Geiger counters sent aboard a rocket. After the satellite Uhuru devoted exc... | {
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The central accreting object is believed to be a black hole rather than a neutron star, since its estimated mass is well above what would be the neutron star mass limit based on any reasonable equation of state.
#### **Exercises**
- **5.1** Derive the general expression (5.1) for pressure in a gas by considering a ... | {
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When we look around at the night sky, we find that the stars are not distributed very uniformly. There is a faint band of light – the Milky Way – going around the celestial sphere in a great circle. Even a moderate telescope reveals that the Milky Way is a collection of innumerable faint stars. [Herschel](#page-476-23)... | {
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In [§3.6.2](#page-105-3) we have discussed globular clusters, which are compact spherical clusters of typically about 106 stars. [Shapley](#page-480-13) [\(1918\)](#page-480-13) noted that most of the globular clusters are found around the constellation Sagittarius in the sky. [Shapley](#page-480-14) [\(1919\)](#page-4... | {
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In other words, the amount of dimming of visible light with distance is approximately equal to 1.5 magnitude kpc−<sup>1</sup> in the galactic plane. The subscript *V* in [\(6.7\)](#page-178-0) implies that we are considering the extinction *A*<sup>λ</sup> in the *V* band introduced in [§1.4.](#page-27-4)
Since the du... | {
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The gravitational field at a point inside or near the Galaxy is expected to be directed towards the galactic centre. How is this gravitational field balanced, to ensure that there is not a general fall of everything towards the galactic centre? There are basically two ways of balancing gravity. A star may move in a cir... | {
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\tag{6.16}$$
Substituting (6.16), we get from (6.13) that
$$v_{\rm R} = \frac{1}{2} \left[ \frac{\Theta_0}{R_0} - \left( \frac{d\Theta}{dR} \right)_{R_0} \right] d \sin 2l, \tag{6.17}$$
whereas (6.14) gives
$$v_{\rm T} = \left[\frac{\Theta_0}{R_0} - \left(\frac{d\Theta}{dR}\right)_{R_0}\right] d\cos^2 l - \frac... | {
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Since the speed $\Theta$ in the $\theta$ direction is given by $\Theta = r\dot{\theta}$ , the above two equations can be written as
$$\ddot{r} = \frac{\Theta^2}{r} - \frac{\Theta_{\text{circ}}^2}{r},\tag{6.26}$$
$$r\Theta = R_0\Theta_0, (6.27)$$
using (6.25) to substitute for $f_r$ and noting that the angu... | {
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\tag{6.39}$$
Here *Z* is the component of velocity perpendicular to the galactic plane and ... implies averaging over stars in the solar neighbourhood. We have shown in [§6.3.1](#page-185-5) that for a particular star varies sinusoidally. The reader is asked in [Exercise 6.4](#page-77-4) to show the same for *Z*. So ... | {
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It was proposed by Schwarzschild (1907) that the stars in the solar neighbourhood would have an ellipsoidal distribution in the velocity space. In other words, the number of stars with velocity components lying between $\Pi$ and $\Pi + d\Pi$ , $\Theta$ and $\Theta + d\Theta$ , Z and Z + dZ should be
$$f(\Pi,\... | {
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We have already pointed out in [§6.2](#page-179-2) that our Galaxy contains two subsystems. One subsystem consists of stars in the disk which revolve around the galactic centre in nearly circular orbits. We shall see in [§6.5](#page-193-2) that the interstellar matter also revolves around the galactic centre with these... | {
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We have seen in [§6.1.3](#page-176-2) that the existence of interstellar matter was established from the extinction and reddening of starlight produced by the interstellar dust. There was evidence that the interstellar space contained much more matter in the form of gas rather than in the form of dust. For example, evi... | {
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In [§2.7](#page-72-1) we discussed how one can analyse a spectral line which is formed by the passage of radiation through an absorbing medium. However, when we analyse radiation that has been emitted by the ISM or that has passed through the ISM, we need to keep in mind that the ISM is far from thermodynamic equilibri... | {
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The relations amongst the Einstein transition coefficients are always given by [\(6.56\).](#page-198-3) Under the condition of thermodynamic equilibrium, *U*<sup>ν</sup> follows from [\(2.6\)](#page-44-2) and the Boltzmann relation [\(6.57\)](#page-198-4) holds. On making use of them, we find that [\(6.59\)](#page-199-... | {
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\tag{6.65}$$
Since $h\nu_0 \ll \kappa_B T$ for the 21-cm line, we get
$$\alpha_{\nu} = \frac{h\nu_0}{c} n_l B_{lu} \frac{h\nu_0}{\kappa_{\rm B} T} \phi(\Delta \nu).$$
Making use of (6.56) and (6.62), this becomes
$$\alpha_{\nu} = \frac{3}{32\pi} n_{\rm H} A_{ul} \frac{hc^2}{\nu_0 \kappa_{\rm B} T} \phi(\Delta... | {
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We pointed out above that the cores of molecular clouds collapse to produce stars. Once the stars have been formed, the UV photons from the O and B stars ionize the ISM around them. Such regions of ionized hydrogen are called *H*II *regions*. The typical temperatures of such regions are of order 6000 K.
![](_page_205... | {
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There is a large-scale magnetic field in the interstellar space of our Galaxy. This was first inferred when [Hiltner](#page-476-29) [\(1954](#page-476-29)) was measuring the polarization of starlight and found that the light from most stars is slightly polarized. It is believed that interstellar grains are generally no... | {
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It should be clear to any reader by now that the ISM is an immensely complex system. Why does the ISM have several different phases in different regions of space instead of being a more uniformly spread gas? We, of course, know that the ISM is continuously disturbed by external sources. Supernova explosions keep adding... | {
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**6.1** We have presented a very elementary discussion of star count analysis in [§6.1.1](#page-172-2) by assuming that all stars have the same absolute magnitude *M* and there is no absorption in interstellar space. Now assume that a fraction of stars (*M*) *dM* have absolute magnitudes between *M* and *M* + *dM*, whe... | {
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Since gravity is a long-range attractive force, any star in a galaxy attracts all the other stars in the galaxy all the time. For simplicity, we can regard the stars as point particles. Then a galaxy or a star cluster can be regarded as a collection of particles in which all the particles are attracting each other thro... | {
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Hence the left-hand side of [\(7.3\)](#page-218-4) must be zero, leading to
$$\sum_{i} \overline{\mathbf{F}_{i}.\mathbf{x}_{i}} + 2\overline{T} = 0. \tag{7.4}$$
Now the force on the *i*-th particle due to all the other particles is
$$\mathbf{F}_i = \sum_{j \neq i} Gm_i \frac{m_j}{|\mathbf{x}_j - \mathbf{x}_i|^3} ... | {
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After establishing the virial theorem which should be valid for any gravitationally bound stellar system in steady state (i.e. which is not growing or shrinking in size), irrespective of whether the system is collisionally relaxed or not, we now come to the important question of estimating the collisional relaxation ti... | {
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If a stellar system has lasted for enough time for collisional relaxation to take place, what should it relax to? This is a question much more difficult to answer than what would appear at the first sight. We know the answer to the similar question for a gas in a container. No matter what initial velocity distribution ... | {
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However, [Chandrasekhar](#page-473-22) [\(1943](#page-473-22)) derived the famous result that a star moving through a stellar system should encounter a drag opposing its motion, giving rise to a frictional term in the evolution equation. Let us qualitatively explain why this should be so. Suppose a star has moved from ... | {
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After discussing the difficulty of obtaining realistic solutions of collisional stellar systems, let us now look at collisionless stellar systems. Since different initial conditions may produce different types of collisionless stellar systems, we would not expect to obtain unique models of such systems from basic princ... | {
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\end{split}$$
On making use of these equations, (7.24) can be written as
$$\frac{\partial f}{\partial t} + \Pi \frac{\partial f}{\partial r} + \frac{\Theta}{r} \frac{\partial f}{\partial \theta} + Z \frac{\partial f}{\partial z} + \left(\frac{\Theta^2}{r} - \frac{\partial \Phi}{\partial r}\right) \frac{\partial f}{... | {
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\tag{7.32}$$
We now have to take the average of this as defined by [\(7.27\).](#page-229-5) It is trivial to see that *Z* = 0 if the distribution function is given by [\(7.31\).](#page-230-2) Keeping in mind that sin α ≈ *z*/*r* and cos α ≈ 1 for small α, [\(7.32\)](#page-231-3) gives
$$\langle \Pi Z \rangle \appro... | {
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In other words, in the absence of random motions, stars have to move in circular orbits with speed $\Theta_{\rm circ}$ in order to be in a steady state. Only when some amount of random motion is present in a group of stars, is it possible for the group to go around the galactic centre with an average speed $\langle ... | {
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As we pointed out in §6.2 and §6.4, our Galaxy has two subsystems. Objects in the first subsystem revolve around the galactic centre in nearly circular orbits, whereas objects in the second subsystem have very low general rotation and are principally balanced against gravity by random motions. Most of the stars in the ... | {
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"Header 2": "7.7 Stars in the solar neighbourhood belonging to two subsystems",
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- **7.1** Suppose *K*(**x**, **v**) is a constant of motion as a star moves around within a stellar system (it can be energy or angular momentum). Show that a distribution function of the form *f* (*K*(**x**, **v**)) will give a time-independent solution of the collisionless Boltzmann equation. This result is known as ... | {
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"Header 2": "**Exercises**",
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A plasma is a gas in which at least some atoms have been broken into positively charged ions and negatively charged electrons. Most of the matter in the Universe exists in the plasma state. The gases inside stars are ionized because of the high temperature, as can be shown easily with the help of the Saha [equation](#p... | {
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"Header 2": "**8.1 Introduction**",
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Our aim is to develop a dynamical theory of fluids, with which we can study how a fluid configuration evolves with time. Any dynamical theory has two requirements. Firstly, we need some means by which we can mathematically prescribe the state of the system at any particular instant of time. Secondly, we need some equat... | {
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If an element of the gas moves under adiabatic conditions, a well-known perfect gas relation implies that $P/\rho^{\gamma}$ will remain invariant for this element, where $\gamma$ is the adiabatic index. Mathematically this can be expressed as
$$\frac{d}{dt}\left(\frac{P}{\rho^{\gamma}}\right) = 0. \tag{8.10}$$
... | {
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We believe that stars form out of the interstellar medium. Star formation is an extremely complex and still ill-understood phenomenon. It is initiated by a fluid dynamical process known as the *Jeans instability*, which breaks the initially uniform interstellar medium into clumps.
Suppose we initially have a uniforml... | {
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"Header 2": "8.3 Jeans instability",
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\tag{8.16}$$
We now have to linearize the Euler [equation \(8.9\),](#page-242-4) which becomes
$$(\rho_0 + \rho_1) \left[ \frac{\partial \mathbf{v}_1}{\partial t} + (\mathbf{v}_1 \cdot \nabla) \mathbf{v}_1 \right] = -\nabla (P_0 + P_1) - (\rho_0 + \rho_1) \nabla (\Phi_0 + \Phi_1).$$
Using (8.12) to cancel two ter... | {
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After familiarizing ourselves with the basic equations of fluid mechanics, let us now consider how these equations have to be generalized to MHD, which essentially treats fluids which are good conductors of electricity. We pointed out at the beginning of [§8.2](#page-239-1) that the state of a neutral fluid can be pres... | {
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Suppose the magnetic field inside the plasma has the typical value *B* and the velocity field has the typical value *V*, whereas *L* is the typical length scale over which the magnetic or velocity fields vary significantly. Then the term ∇ × (**v** × **B**) in the induction [equation \(8.27\)](#page-248-3) should be of... | {
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Hence we have
$$\int_{S} \mathbf{B} \cdot \frac{d}{dt} (d\mathbf{S}) = \oint_{C} (\mathbf{B} \times \mathbf{v}) \cdot \delta \mathbf{l} = \int_{S} [\nabla \times (\mathbf{B} \times \mathbf{v})] \cdot d\mathbf{S}$$
by Stokes's theorem. We then have from (8.35) that
$$\frac{d}{dt} \int_{S} \mathbf{B} . d\mathbf{S} ... | {
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