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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 ... | {
"Header 2": "**Astrophysics for Physicists**",
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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... | {
"Header 1": "**Astrophysics for Physicists**",
"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... | {
"Header 1": "**Interaction of radiation with matter**",
"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... | {
"Header 1": "**Interaction of radiation with matter**",
"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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"Header 2": "**2.4 Radiative transfer through stellar atmospheres**",
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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
$$I_{\nu}(\tau_{\nu}, \mu) = B_{\nu}(\tau... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.4 Radiative transfer through stellar atmospheres**",
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Just similar to (2.39), (2.40) and (2.41), we write down the total energy density, total radiation flux and total radiation pressure integrated over all frequencies:
$$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... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.4 Radiative transfer through stellar atmospheres**",
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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... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.4 Radiative transfer through stellar atmospheres**",
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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 ... | {
"Header 1": "**Interaction of radiation with matter**",
"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 ... | {
"Header 1": "**Interaction of radiation with matter**",
"Header 2": "**2.8 Photon diffusion inside the Sun**",
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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... | {
"Header 1": "**Stellar astrophysics I: Basic theoretical ideas and observational data**",
"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*... | {
"Header 1": "**Stellar astrophysics I: Basic theoretical ideas and observational data**",
"Header 2": "**3.2 Basic equations of stellar structure**",
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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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"Header 2": "**3.2 Basic equations of stellar structure**",
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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... | {
"Header 1": "**Stellar astrophysics I: Basic theoretical ideas and observational data**",
"Header 2": "**3.3 Constructing stellar models**",
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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... | {
"Header 1": "**Stellar astrophysics I: Basic theoretical ideas and observational data**",
"Header 2": "**3.3 Constructing stellar models**",
"token_count": 944,
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