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In reality, the evolution of our universe is complicated by the fact that it contains different components with different equations of state. We know that the universe contains non-relativistic matter and radiation – that's a conclusion as firm as the earth under your feet and as plain as daylight. Thus, the universe c... | {
"Header 1": "Single-Component Universes",
"Header 2": "5.1 Evolution of energy density",
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A detailed calculation indicates that the energy density of each neutrino flavor should be
$$\varepsilon = \frac{7}{8} \left(\frac{4}{11}\right)^{4/3} \varepsilon_{\text{CMB}} \approx 0.227 \varepsilon_{\text{CMB}} \ .$$
(5.16)
(The above result assumes that the neutrinos are relativistic, or, equivalently, that th... | {
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"Header 2": "5.1 Evolution of energy density",
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\tag{5.27}$$
Each term on the right hand side of equation (5.27) has a different dependence on scale factor; radiation contributes a term $\propto a^{-2}$ , matter contributes a term $\propto a^{-1}$ , curvature contributes a term independent of a, and the cosmological constant $\Lambda$ contributes a term $\pro... | {
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"Header 2": "5.1 Evolution of energy density",
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In such a spatially flat, single-component universe, the Friedmann equation takes the simple form
$$\dot{a}^2 = \frac{8\pi G \varepsilon_0}{3c^2} a^{-(1+3w)} \ . \tag{5.39}$$
To solve this equation, we first make the educated guess that the scale factor has the power law form $a \propto t^q$ . The left hand side o... | {
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"Header 2": "5.1 Evolution of energy density",
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Let's now look at specific examples of spatially flat universes, starting with a universe containing only non-relativistic matter (w = 0).<sup>5</sup> The age of such a universe is
$$t_0 = \frac{2}{3H_0} \,\,\,(5.57)$$
and the horizon distance is
$$d_{\text{hor}}(t_0) = 3ct_0 = 2c/H_0 . (5.58)$$
The scale facto... | {
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"Header 2": "5.4 Matter only",
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The case of a spatially flat universe containing only radiation is of particular interest, since early in the history of our own universe, the radiation (w = 1/3) term dominated the right-hand side of the Friedmann equation (see equation 5.27). Thus, at early times – long before the time of radiationmatter equality – t... | {
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"Header 2": "5.5 Radiation only",
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As seen in Section 5.3, a spatially flat, single-component universe with $w \neq -1$ has a power-law dependence of scale factor on time:
$$a \propto t^{2/(3+3w)}$$
(5.74)
Now, for the sake of completeness, consider the case with w = -1; that is, a universe in which the energy density is contributed by a cosmologi... | {
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"Header 3": "5.6 Lambda only",
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[Full references are given in the "Annotated Bibiography" on page 286.]
Liddle (1999), ch. 4: Flat universes, both matter-only and radiation-only
Linder (1997), ch. 2.4,2.5: Evolution of energy density; evolution of scale factor in single-component universes
#### **Problems**
(5.1) The predicted number of neutr... | {
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"Header 2": "Suggested reading",
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The Friedmann equation, in general, can be written in the form
$$H(t)^{2} = \frac{8\pi G}{3c^{2}}\varepsilon(t) - \frac{\kappa c^{2}}{R_{0}^{2}a(t)^{2}},$$
(6.1)
where H ≡ a/a ˙ , and ε(t) is the energy density contributed by all the components of the universe, including the cosmological constant. Equation (4.31) t... | {
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Eliot, Frost was keenly interested in astronomy, and frequently wrote poems on astronomical themes.
Figure 6.1: The scale factor as a function of time for universes containing only matter. The dotted line is a(t) for a universe with Ω<sup>0</sup> = 1 (flat); the dashed line is a(t) for ... | {
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"Header 2": "6.1 Matter + curvature",
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Consider a universe which is spatially flat, but which contains both matter and a cosmological constant.<sup>4</sup> If, at a given time t = t0, the density parameter
<sup>4</sup>Such a universe is of particular interest to us, since it appears to be a close approximation to our own universe at the present day.
in ... | {
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"Header 2": "6.2 Matter + lambda",
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(We'll see in section 6.5 that ignoring the radiation content of the universe has an insignificant effect on our estimate of $t_0$ .) The age at which matter and the cosmological constant had equal energy density was
$$t_{m\Lambda} = \frac{2H_0^{-1}}{3\sqrt{1 - \Omega_{m,0}}} \ln[1 + \sqrt{2}] = 0.702H_0^{-1} = 9.8 ... | {
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This universe lies above the Big Chill – Big Bounce dividing line in Figure 6.3; it is a positively curved universe which "bounced" at a scale factor $a=a_{\text{bounce}}\approx 0.56$ .
There is strong observational evidence that we do not live in a loitering or Big Bounce universe. If we lived in a loitering univer... | {
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"Header 2": "6.2 Matter + lambda",
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In our universe, radiation-matter equality took place at a scale factor $a_{rm} \equiv \Omega_{r,0}/\Omega_{m,0} \approx 2.8 \times 10^{-4}$ . At scale factors $a \ll a_{rm}$ , the universe is well described by a flat, radiation-only model, as described in Section 5.5. At scale factors $a \sim a_{rm}$ , the universe... | {
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"Header 2": "6.2 Matter + lambda",
"Header 3": "6.4 Radiation + matter",
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The current age of the universe, in the Benchmark Model, is $t_0 = 13.5 \,\mathrm{Gyr}$ .
With $\Omega_{r,0}$ , $\Omega_{m,0}$ , and $\Omega_{\Lambda,0}$ known, the scale factor a(t) can be computed numerically using the Friedmann equation, in the form of equation (6.6). Figure 6.5 shows the scale factor, thus c... | {
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"Header 2": "6.2 Matter + lambda",
"Header 3": "6.4 Radiation + matter",
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(6.1) In a positively curved universe containing only matter (Ω<sup>0</sup> > 1, κ = +1), show that the present age of the universe is given by the formula
$$H_0 t_0 = \frac{\Omega_0}{2(\Omega_0 - 1)^{3/2}} \cos^{-1} \left(\frac{2 - \Omega_0}{\Omega_0}\right) - \frac{1}{\Omega_0 - 1} . \tag{6.43}$$
Assuming H<sup>0... | {
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"Header 2": "Problems",
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Since determining the exact functional form of a(t) is difficult, it is useful, instead, to do a Taylor series expansion for a(t) around the present moment. The complete Taylor series is
$$a(t) = a(t_0) + \frac{da}{dt} \bigg|_{t=t_0} (t - t_0) + \frac{1}{2} \left. \frac{d^2a}{dt^2} \right|_{t=t_0} (t - t_0)^2 + \dots... | {
"Header 1": "Measuring Cosmological Parameters",
"Header 3": "7.1 \"A search for two numbers\"",
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If we observe, at time t0, light that was emitted by a distant galaxy at time te, the
<sup>2</sup>The peculiar velocities of galaxies cause a significant amount of scatter in the plot, but by using a large number of galaxies, you can beat down the statistical errors. If you use galaxies at d < 100 Mpc, you must also ... | {
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Unfortunately, the current proper distance to a galaxy, $d_p(t_0)$ , is not a measurable property. If you tried to measure the distance to a galaxy with a tape measure, for instance, the distance would be continuously increasing as you extended the tape. To measure the proper distance at time $t_0$ , you would need a... | {
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"Header 3": "7.2 Luminosity distance",
"token_count": 1998,
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The net result is that in an expanding, spatially curved universe, the relation between the observed flux f and the luminosity L of a distant light source is
$$f = \frac{L}{4\pi S_{\kappa}(r)^2 (1+z)^2} , \qquad (7.27)$$
and the luminosity distance is
$$d_L = S_{\kappa}(r)(1+z) . \tag{7.28}$$
![](_page_140_Fi... | {
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The luminosity distance d<sup>L</sup> is not the only distance measure that can be computed using the observable properties of cosmological objects. Suppose that instead of a standard candle, you observed a standard yardstick. A standard yardstick is an object whose proper length ` is known. In most cases, it is conven... | {
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For the Benchmark Model,
$$\delta\theta(\min) = \frac{\ell}{d_A(\max)} = \frac{\ell}{1800 \,\text{Mpc}} \approx 0.1 \,\text{arcsec}\left(\frac{\ell}{1 \,\text{kpc}}\right) \,.$$
(7.42)
Both galaxies and clusters of galaxies are large enough to be useful standard candles. Unfortunately for cosmologists, galaxies and... | {
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"Header 2": "7.3 Angular-diameter distance",
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Observation of Cepheid stars in the Virgo cluster of galaxies, for instance, has yielded a distance $d_L(\mathrm{Virgo}) = 300\,d_L(\mathrm{LMC}) = 15\,\mathrm{Mpc}$ . One of the motivating reasons for building the Hubble Space Telescope in the first place was to use Cepheids to determine $H_0$ . The net result of th... | {
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To determine the value of H<sup>0</sup> without having to worry about Virgocentric flow and other peculiar velocities, we need to determine the luminosity distance to standard candles with d<sup>L</sup> > 100 Mpc, or z > 0.02. To determine the value of q0, we need to view standard candles for which the relation between... | {
"Header 1": "Measuring Cosmological Parameters",
"Header 2": "7.5 Standard candles & the accelerating universe",
"token_count": 1995,
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Nowadays, the bolometric apparent magnitude of a light source is defined in terms of the source's bolometric flux as
$$m \equiv -2.5 \log_{10}(f/f_x)$$
, (7.46)
where the reference flux <sup>f</sup><sup>x</sup> is set at the value <sup>f</sup><sup>x</sup> <sup>=</sup> <sup>2</sup>.<sup>53</sup> <sup>×</sup> <sup>... | {
"Header 1": "Measuring Cosmological Parameters",
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Note that decelerating universes (with $q_0 > 0$ ) can be strongly excluded by the data, as can Big Crunch universes (labeled 'Recollapses' in Figure 7.6), and Big Bounce universes (labeled 'No Big Bang' in Figure 7.6). The supernova data are consistent with negative curvature (labeled 'Open' in Figure 7.6), positive ... | {
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"Header 2": "7.5 Standard candles & the accelerating universe",
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(7.1) Suppose that a polar bear's foot has a luminosity of L = 10 watts. What is the bolometric absolute magnitude of the bear's foot? What is the bolometric apparent magnitude of the foot at a luminosity distance of d<sup>L</sup> = 0.5 km? If a bolometer can detect the bear's foot at a maximum luminosity distance of d... | {
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"Header 2": "Problems",
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Cosmologists, over the years, have dedicated a large amount of time and effort to determining the matter density of the universe. There are many reasons for this obsession. First, the density parameter in matter, $\Omega_{m,0}$ , is important in determining the spatial curvature and expansion rate of the universe. Eve... | {
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"Header 2": "Dark Matter",
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If the mass-to-light ratio of the stars in the Coma cluster is $\langle M/L_B \rangle \approx 4 \, \rm{M_{\odot}/\,L_{\odot,B}}$ , then the total mass of stars in the Coma cluster is $M_{\text{Coma},\star} \approx 3 \times 10^{13} \,\mathrm{M}_{\odot}$ . Although 30 trillion solar masses represents a lot of stars, th... | {
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The open circles show the results of Rubin and Ford (1970, ApJ, 159, 379) at visible wavelengths; the solid dots with error bars show the results of Roberts and Whitehurst (1975, ApJ, 201, 327) at radio wavelengths (figure from van den Bergh, 2000).
The first astronomer to detect the rotation of M31 was Vesto Slipher... | {
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The acceleration of the
<sup>&</sup>lt;sup>5</sup>Although Zwicky's work popularized the phrase "dark matter", he was not the first to use it in an astronomical context. For instance, in 1908, Henri Poincaré discussed the possible existence within our Galaxy of "matière obscure" (translated as "dark matter" in the st... | {
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"Header 2": "Dark Matter",
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From measurements of the redshifts of hundreds of galaxies in the Coma cluster, the mean redshift of the cluster is found to be
$$\langle z \rangle = 0.0232 \;, \tag{8.35}$$
which can be translated into a radial velocity
$$\langle v_r \rangle = c \langle z \rangle = 6960 \,\mathrm{km \, s^{-1}}$$
(8.36)
and a d... | {
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"Header 2": "Dark Matter",
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We can detect dark matter in clusters of galaxies because it affects the motions of
<sup>&</sup>lt;sup>7</sup>The roots of these methods can be traced back as far as the year 1846, when Leverrier and Adams deduced the existence of the dim planet Neptune by its effect on the orbit of Uranus.
![](_page_176_Picture_2.... | {
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The elongated arcs seen in Figure 8.7 are not oddly shaped galaxies within the cluster; instead, they are background galaxies, at redshifts z > 0.18, which are gravitationally lensed by the cluster mass. The mass of clusters can be estimated by the degree to which they lens background galaxies. The masses calculated in... | {
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"Header 2": "Dark Matter",
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(8.1) Suppose it were suggested that black holes of mass 10<sup>−</sup><sup>8</sup> M¯ made up all the dark matter in the halo of our Galaxy. How far away would you expect the nearest such black hole to be? How frequently would you expect such a black hole to pass within 1 AU of the Sun? (An order-of-magnitude estimate... | {
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"Header 2": "Problems",
"token_count": 895,
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If Heinrich Olbers had lived in intergalactic space and had eyes that operated at millimeter wavelengths (admittedly a very large "if"), he would not have formulated Olbers' Paradox. At wavelengths of a few millimeters, thousands of times longer than human eyes can detect, most of the light in the universe comes not fr... | {
"Header 1": "The Cosmic Microwave Background",
"token_count": 807,
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Although CMB photons are as common as dirt,<sup>1</sup>Arno Penzias and Robert Wilson were surprised when they serendipitously discovered the Cosmic Microwave Background. At the time of their discovery, Penzias and Wilson were radio astronomers working at Bell Laboratories. The horn-reflector radio antenna which they u... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.1 Observing the CMB",
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\tag{9.7}$$
<sup>&</sup>lt;sup>4</sup>The dipole distortion of the CMB was first detected in 1977, using aircraft-borne and balloon-borne detectors. The unique contribution of COBE was the precision with which it measured the temperature distortion.
<sup>&</sup>lt;sup>5</sup>The distorted "yin-yang" pattern in the ... | {
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"Header 2": "9.1 Observing the CMB",
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To understand in more detail the origin of the Cosmic Microwave Background, we'll have to examine fairly carefully the process by which the baryonic matter goes from being an ionized plasma to a gas of neutral atoms, and the closely related process by which the universe goes from being opaque to being transparent. To a... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.2 Recombination and decoupling",
"token_count": 1990,
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During the early stages of the universe ( $a < a_{rm} \approx 3 \times 10^{-4}$ ) the universe was radiation dominated, and the Friedmann equation was
$$\frac{H^2}{H_0^2} = \frac{\Omega_{r,0}}{a^4} \ . \tag{9.17}$$
Thus, the Hubble parameter was
$$H = \frac{H_0 \Omega_{r,0}^{1/2}}{a^2} = \frac{2.1 \times 10^{-20}... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.2 Recombination and decoupling",
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When does recombination, and the consequent photon decoupling, take place? It's easy to do a quick and dirty approximation of the recombination temperature. Recombination, one could argue, must take place when the mean
energy per photon of the Cosmic Microwave Background falls below the ionization energy of hydrogen,... | {
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"Header 2": "9.2 Recombination and decoupling",
"Header 3": "9.3 The physics of recombination",
"token_count": 1976,
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In degrees Kelvin, kTrec = 0.323 eV corresponds to a temperature Trec = 3740 K, slightly higher than the melting point of tungsten.<sup>10</sup> The temperature of the universe had a value T = Trec = 3740 K at a redshift zrec = 1370, when the age of the universe, in the Benchmark Model, was trec = 240,000 yr. Recombina... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.2 Recombination and decoupling",
"Header 3": "9.3 The physics of recombination",
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Since the photons traveled about at the speed of light, kicking the electrons
<sup>11</sup>By the time the universe becomes Λ dominated, the free electron density has fallen to negligibly small levels, so using the Hubble parameter for a matter-dominated universe is a justifiable approximation in computing τ .
befo... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.2 Recombination and decoupling",
"Header 3": "9.3 The physics of recombination",
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The dipole distortion of the Cosmic Microwave Background, shown in the top panel of Figure 9.2, results from the fact that the universe is not perfectly homogeneous today (at z = 0). Because we are gravitationally accelerated towards the nearest large lumps of matter, we see a Doppler shift in the radiation of the CMB.... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.4 Temperature fluctuations",
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Note that the temperature fluctuation has a peak at l ∼ 200, corresponding to an angular size of ∼ 1 ◦ . The detailed shape of the ∆<sup>T</sup> versus l curve, as shown in Figure 9.6, contains a wealth of information about the universe at the time of photon decoupling. In the next section, we will examine, very briefl... | {
"Header 1": "The Cosmic Microwave Background",
"Header 2": "9.4 Temperature fluctuations",
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If the photon-baryon fluid is in the process of expanding or contracting at the time of decoupling, the Doppler effect will cause the liberated photons to be cooler or hotter than average, depending on whether the photon-baryon fluid was moving away from our location or toward it at the time of photon decoupling. Compu... | {
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"Header 2": "9.4 Temperature fluctuations",
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The Cosmic Microwave Background tells us a great deal about the state of the universe at the time of last scattering ( $t_{\rm ls} \approx 0.35\,{\rm Myr}$ ). However, the opacity of the early universe prevents us from directly seeing what the universe was like at $t < t_{\rm ls}$ . Photons are the "messenger boys" of... | {
"Header 1": "Nucleosynthesis & the Early Universe",
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That is, deuterium synthesis occurred at a temperature $T_{\rm nuc}\approx 1.6\times10^5(3740\,{\rm K})\approx 6\times10^8\,{\rm K}$ , corresponding to a time $t_{\rm nuc}\approx 300\,{\rm s}$ . This estimate, as we'll see when we do the detailed calculations, gives a temperature slightly too low, but it certainly gi... | {
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} |
the interaction of a neutrino with any other particle via the weak nuclear force is
$$\sigma_w \sim 10^{-47} \,\mathrm{m}^2 \left(\frac{kT}{1 \,\mathrm{MeV}}\right)^2 \ .$$
(10.15)
(Compare this to the Thomson cross-section for the interaction of electrons via the electromagnetic force: $\sigma_e = 6.65 \times 1... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"token_count": 2009,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
Let's move on to the next stage of Big Bang Nucleosynthesis, just after proton-neutron freezeout is complete. The time is t ≈ 2 s. The neutron-toproton ratio is nn/n<sup>p</sup> = 0.2. The neutrinos, which ceased to interact with electrons about the same time they stopped interacting with neutrons and protons, are now ... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"Header 2": "10.3 Deuterium synthesis",
"token_count": 1923,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
With $m_n c^2 = 939.6 \,\mathrm{MeV}$ , $B_D = 2.22 \,\mathrm{MeV}$ , and $\eta = 5.5 \times 10^{-10}$ , the temperature of deuterium synthesis is $kT_{\mathrm{nuc}} \approx 0.066 \,\mathrm{MeV}$ , corresponding to $T_{\mathrm{nuc}} \approx 7.6 \times 10^8 \,\mathrm{K}$ . The temperature drops to this value when... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"Header 2": "10.3 Deuterium synthesis",
"token_count": 2027,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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Small amounts of D, <sup>3</sup>H, and <sup>3</sup>He are left over, a tribute to the incomplete nature of Big Bang Nucleosynthesis. (The <sup>3</sup>H later decays to <sup>3</sup>He.) Very small amounts of <sup>6</sup>Li, <sup>7</sup>Li, and <sup>7</sup>Be are made. (The <sup>7</sup>Be is later converted to <sup>7</su... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"Header 2": "10.3 Deuterium synthesis",
"token_count": 1942,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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The value of η can be converted into a value for the current baryon density by the relation
$$n_{\text{bary},0} = \eta n_{\gamma,0} = 0.23 \pm 0.02 \,\text{m}^{-3} \ .$$
(10.43)
Since most of the baryons are protons, we can write, to acceptable accuracy,
$$\varepsilon_{\text{bary},0} = (m_p c^2) n_{\text{bary},... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"Header 2": "10.3 Deuterium synthesis",
"token_count": 345,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The results of Big Bang Nucleosynthesis tell us what the universe was like when it was relatively hot (Tnuc ≈ 7 × 10<sup>8</sup> K) and dense:
$$\varepsilon_{\rm nuc} \approx \alpha T_{\rm nuc}^4 \approx 10^{33} \,\mathrm{MeV} \,\mathrm{m}^{-3} \ .$$
(10.46)
This energy density corresponds to <sup>a</sup> mass dens... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"Header 2": "10.5 Baryon – antibaryon asymmetry",
"token_count": 1944,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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With these assumptions, what was the energy density ε at the time of nucleosynthesis? What was the Hubble parameter H at the time of nucleosynthesis? What was the time tnuc at which nucleosynthesis took place? What is the current temperature T<sup>0</sup> of the radiation filling the universe today? If the universe swi... | {
"Header 1": "Nucleosynthesis & the Early Universe",
"Header 2": "10.5 Baryon – antibaryon asymmetry",
"token_count": 677,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The observed properties of galaxies, quasars, and supernovae at relatively small redshift (z < 6) tell us about the universe at times $t > 1\,\mathrm{Gyr}$ . The properties of the Cosmic Microwave Background tell us about the universe at the time of last scattering ( $z_{\rm ls} \approx 1100, t_{\rm ls} \approx 0.35\,... | {
"Header 1": "Inflation & the Very Early Universe",
"token_count": 2042,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The "flatness problem", the remarkable closeness of Ω to one in the early universe, is puzzling. It is accompanied, however, by the "horizon problem", which is, if anything, even more puzzling. The "horizon problem" is simply the statement that the universe is nearly homogeneous and isotropic on very large scales. Why ... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "11.2 The horizon problem",
"token_count": 1306,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The monopole problem – that is, the apparent lack of magnetic monopoles in the universe – is not a purely cosmological problem, but one that results from combining the Hot Big Bang scenario with the particle physics concept of a Grand Unified Theory. In particle physics, a Grand Unified Theory, or GUT, is a field theor... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "11.3 The monopole problem",
"token_count": 2045,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
What is inflation? In a cosmological context, inflation can most generally be defined as the hypothesis that there was a period, early in the history of our universe, when the expansion was accelerating outward; that is, an epoch when a¨ > 0. The acceleration equation,
$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^2} (\va... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "11.4 The inflation solution",
"token_count": 2035,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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If inflation started at $t_i \approx 10^{-36} \,\mathrm{s}$ , with a Hubble parameter $H_i \approx 10^{36} \,\mathrm{s}^{-1}$ , and lasted for $n \approx 100$ e-foldings, then the horizon size immediately before inflation was
$$d_{\text{hor}}(t_i) = 2ct_i \approx 6 \times 10^{-28} \,\text{m} \ .$$
(11.37)
The h... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "11.4 The inflation solution",
"token_count": 1961,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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Under what circumstances are the conditions for inflation (small $\dot{\phi}$ and large V) met in the early universe? To determine the value of $\dot{\phi}$ , start with the fluid equation for the energy density of the inflaton field,
$$\dot{\varepsilon}_{\phi} + 3H(t)(\varepsilon_{\phi} + P_{\phi}) = 0 , \qquad... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "11.4 The inflation solution",
"token_count": 2044,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The energy lost by the inflaton field after its phase transition from the false vacuum to the true vacuum can be thought of as the latent heat of that transition. When water freezes, to use a low-energy analogy, it loses an energy of $3 \times 10^8 \, \mathrm{J} \, \mathrm{m}^{-3}$ , which goes to heat its surrounding... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "11.4 The inflation solution",
"token_count": 999,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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[Full references are given in the "Annotated Bibliography" on page 286.]
Islam (2002), ch. 9: A general overview of inflation, avoiding technical concepts of particle physics.
Liddle (1999), ch. 11: A brief, clear discussion of how inflation solves the horizon, flatness, and monopole problems.
Liddle & Lyth (2000... | {
"Header 1": "Inflation & the Very Early Universe",
"Header 2": "Suggested reading",
"token_count": 641,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
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The universe can be approximated as being homogeneous and isotropic only if we smooth it with a filter ∼ 100 Mpc across. On smaller scales, the universe contains density fluctuations ranging from subatomic quantum fluctuations up to the large superclusters and voids, ∼ 50 Mpc across, which characterize the distribution... | {
"Header 1": "The Formation of Structure",
"token_count": 1199,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
To put our study of gravitational instability on a more quantitative basis, consider some component of the universe whose energy density ε(~r,t) is a function of position as well as time. At a given time t, the spatially averaged energy density is
$$\bar{\varepsilon}(t) = \frac{1}{V} \int_{V} \varepsilon(\vec{r}, t) ... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.1 Gravitational instability",
"token_count": 2032,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The steepening of the pressure gradient, however, doesn't occur instantaneously. Any change in pressure travels at the sound speed.<sup>4</sup> Thus, the time it takes for the pressure gradient to build up in a region of radius R will be
$$t_{\rm pre} \sim \frac{R}{c_s} \,, \tag{12.16}$$
where c<sup>s</sup> is the ... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.1 Gravitational instability",
"token_count": 1970,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The sound speed in the photon gas is
$$c_s(\text{photon}) = c/\sqrt{3} \approx 0.58c$$
(12.30)
The sound speed in the baryonic gas, by contrast, is
$$c_s(\text{baryon}) = \left(\frac{kT}{mc^2}\right)^{1/2} c . \qquad (12.31)$$
At the time of decoupling, the thermal energy per particle was kTdec ≈ 0.26 eV, and t... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.1 Gravitational instability",
"token_count": 1999,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
Combining equations (12.38) and (12.42), we find
$$\frac{\ddot{a}}{a} - \frac{1}{3}\ddot{\delta} - \frac{2}{3}\frac{\dot{a}}{a}\dot{\delta} = -\frac{4\pi}{3}G\bar{\rho} - \frac{4\pi}{3}G\bar{\rho}\delta . \qquad (12.43)$$
When δ = 0, equation (12.43) reduces to
$$\frac{\ddot{a}}{a} = -\frac{4\pi}{3}G\bar{\rho} , ... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.1 Gravitational instability",
"token_count": 2035,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
When deriving equation (12.46), which determines the growth rate of density perturbations in a Newtonian universe, I assumed that the perturbation was spherically symmetric. In fact, equation (12.46) and its relativistically correct brother, equation (12.48), both apply to low-amplitude perturbations of any shape. This... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.4 The power spectrum",
"token_count": 2018,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
The mean mass of each sphere (considering only the non-relativistic matter which it contains) will be
$$\langle M \rangle = \frac{4\pi}{3} L^3 \frac{\varepsilon_{m,0}}{c^2} \ . \tag{12.71}$$
However, the actual mass of each sphere will vary; some spheres will be slightly underdense, and others will be slightly over... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.4 The power spectrum",
"token_count": 585,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
Immediately after inflation, the expected power spectrum for density perturbations has the form P(k) ∝ k n , with an index n = 1 being predicted by most inflationary models. However, the shape of the power spectrum will be modified between the end of inflation at t<sup>f</sup> and the time of radiation-matter equality ... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.5 Hot versus cold",
"token_count": 2009,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
Remember, when the universe is radiation-dominated, density fluctuations $\delta_{\vec{k}}$ in the dark matter do not grow appreciably in amplitude, as long as their proper wavelength $a(t)2\pi/k$ is small compared to the Hubble distance c/H(t). However, when the proper wavelength of a density perturbation is large... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.5 Hot versus cold",
"token_count": 2049,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
Show that $t_{\rm min} \approx R_{\rm halo}/v$ for a disk galaxy. What is $t_{\rm min}$ for our own Galaxy? What is the maximum possible redshift at which you would expect to see galaxies comparable in v and $R_{\rm halo}$ to our own Galaxy? (Assume the Benchmark Model is correct.)
- (12.5) Within the Coma cluste... | {
"Header 1": "The Formation of Structure",
"Header 2": "12.5 Hot versus cold",
"token_count": 412,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
A book dealing with an active field like cosmology can't really have a neat, tidy ending. Our understanding of the universe is still growing and evolving. During the twentieth century, the growing weight of evidence pointed toward the Hot Big Bang model, in which the universe started in a hot, dense state, but graduall... | {
"Header 1": "Epilogue",
"token_count": 727,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
- Harrison, E. R. 1987, Darkness at Night: A Riddle of the Universe (Cambridge: Harvard University Press) A comprehensive discussion of Olbers' Paradox and its place in the history of cosmology.
- Kragh, H. 1996, Cosmology and Controversy (Princeton: Princeton University Press) A well-reseached history of the Big Bang ... | {
"Header 1": "Annotated Bibliography",
"Header 2": "Popular",
"token_count": 226,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
• Bernstein, J. 1995, Introduction to Cosmology (Englewood Cliffs, NJ: Prentice Hall) Has a slightly greater emphasis on particle physics than most cosmology texts.
- Coles, P. 1999, The Routledge Critical Dictionary of the New Cosmology (New York: Routledge) In addition to a dictionary of cosmology-related terms, fr... | {
"Header 1": "Annotated Bibliography",
"Header 2": "Intermediate",
"token_count": 625,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Ryden_IntroCosmo.pdf"
} |
This introductory chapter sets the stage for the course, and briefly repeats some concepts from earlier courses on stellar astrophysics (e.g. the Utrecht first-year course *Introduction to stellar structure and evolution* by F. Verbunt).
The *goal* of this course on stellar evolution can be formulated as follows:
t... | {
"Header 1": "Introduction",
"token_count": 410,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Fundamental properties of a star include the *mass M* (usually expressed in units of the solar mass, $M_{\odot} = 1.99 \times 10^{33}$ g), the *radius R* (often expressed in $R_{\odot} = 6.96 \times 10^{10}$ cm) and the *luminosity L*, the rate at which the star radiates energy into space (often expressed in $L_{\... | {
"Header 1": "1.1 Observational constraints",
"token_count": 1485,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The Hertzsprung-Russell diagram (HRD) is an important tool to test the theory of stellar evolution. Fig. 1.1 shows the colour-magnitude diagram (CMD) of stars in the vicinity of the Sun, for which the Hipparcos satellite has measured accurate distances. This is an example of a *volume-limited* sample
![](_page_7_Figu... | {
"Header 1": "**1.1.1 The Hertzsprung-Russell diagram**",
"token_count": 705,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Stars in the Galaxy are divided into different populations:
- Population I: stars in the galactic disk, in spiral arms and in (relatively young) open clusters. These stars have ages $\lesssim 10^9$ yr and are relatively metal-rich ( $Z \sim 0.5-1\,Z_\odot$ )
- Population II: stars in the galactic halo and in globul... | {
"Header 1": "1.2 Stellar populations",
"token_count": 333,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We wish to build a theory of stellar evolution to explain the observational constraints highlighted above. In order to do so we must make some basic assumptions:
• stars are considered to be *isolated* in space, so that their structure and evolution depend only on *intrinsic* properties (mass and composition). For mo... | {
"Header 1": "1.3 Basic assumptions",
"token_count": 476,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
#### **1.1 Evolutionary stages**
In this course we use many concepts introduced in the introductory astronomy classes. In this exercise we recapitulate the names of evolutionary phases. During the lectures you are assumed to be familiar with these terms, in the sense that you are able to explain them in general terms... | {
"Header 1": "**Exercises**",
"token_count": 627,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
(a) The masses of stars are approximately in the range 0.08 *M*<sup>⊙</sup> . *M* . 100 *M*⊙. Why is there an upper limit? Why is there a lower limit?
- (b) Can you think of methods to measure (1) the mass, (2) the radius, and (3) the luminosity of a star? Can your methods be applied for any star or do they require s... | {
"Header 1": "**1.3 Mass-luminosity and mass-radius relation**",
"token_count": 505,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The assumption of spherical symmetry implies that all interior physical quantities (such as density ρ, pressure *P*, temperature *T*, etc) depend only on one radial coordinate. The obvious coordinate to use in a Eulerian coordinate system is the radius of a spherical shell, *r* (∈ 0 . . . *R*). In an evolving star, all... | {
"Header 1": "**2.1 Coordinate systems and the mass distribution**",
"token_count": 1025,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Recall that a star is a self-gravitating body of gas, which implies that gravity is the driving force behind stellar evolution. In the general, non-spherical case, the gravitational acceleration *g* can be written as the gradient of the gravitational potential, *<sup>g</sup>* <sup>=</sup> <sup>−</sup>∇Φ, where <sup>Φ</... | {
"Header 1": "**2.1.1 The gravitational field**",
"token_count": 239,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We next consider conservation of momentum inside a star, i.e. Newton's second law of mechanics. The net acceleration on a gas element is determined by the sum of all forces acting on it. In addition to the gravitational force considered above, forces result from the pressure exerted by the gas surrounding the element. ... | {
"Header 1": "**2.2 The equation of motion and hydrostatic equilibrium**",
"token_count": 1681,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We can ask what happens if the state of hydrostatic equilibrium is violated: how fast do changes to the structure of a star occur? The answer is provided by the equation of motion, eq. (2.10). For example, suppose that the pressure gradient that supports the star against gravity suddenly drops. All mass shells are then... | {
"Header 1": "**2.2.1 The dynamical timescale**",
"token_count": 831,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
An important consequence of hydrostatic equilibrium is the *virial theorem*, which is of vital importance for the understanding of stars. It connects two important energy reservoirs of a star and allows predictions and interpretations of important phases in the evolution of stars.
To derive the virial theorem we star... | {
"Header 1": "2.3 The virial theorem",
"token_count": 1917,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The total energy of a star is the sum of its gravitational potential energy, its internal energy and its kinetic energy *E*kin (due to bulk motions of gas inside the star, not the thermal motions of the gas particles):
$$E_{\text{tot}} = E_{\text{gr}} + E_{\text{int}} + E_{\text{kin}}. \tag{2.32}$$
The star is boun... | {
"Header 1": "**2.3.1 The total energy of a star**",
"token_count": 786,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
If internal energy sources are present in a star due to nuclear reactions taking place in the interior, then the energy loss from the surface can be compensated: $L = L_{\text{nuc}} \equiv -dE_{\text{nuc}}/dt$ . In that case the total energy is conserved and eq. (2.34) tells us that $\dot{E}_{\text{tot}} = \dot{E}_{\... | {
"Header 1": "2.3.2 Thermal equilibrium",
"token_count": 474,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Three important timescales are relevant for stellar evolution, associated with changes to the mechanical structure of a star (described by the equation of motion, eq. 2.11), changes to its thermal structure (as follows from the virial theorem, see also Sect. 5.1) and changes in its composition, which will be discussed ... | {
"Header 1": "2.3.2 Thermal equilibrium",
"Header 2": "2.4 The timescales of stellar evolution",
"token_count": 1297,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The contents of this chapter are covered more extensively by Chapter 1 of Maeder and by Chapters 1 to 4 of Kippenhahn & Weigert.
#### **Exercises**
#### 2.1 Density profile
In a star with mass M, assume that the density decreases from the center to the surface as a function of radial distance r, according to
$$... | {
"Header 1": "Suggestions for further reading",
"token_count": 1339,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
- (a) The nuclear timescale τnuc.
- i. Calculate the total mass of hydrogen available for fusion over the lifetime of the Sun, if 70% of its mass was hydrogen when the Sun was formed, and only 13% of all hydrogen is in the layers where the temperature is high enough for fusion.
- ii. Calculate the fractional amount of ... | {
"Header 1": "Suggestions for further reading",
"Header 2": "**2.5 Three important timescales in stellar evolution**",
"token_count": 630,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Empirical evidence shows that in a part of space isolated from the rest of the Universe, matter and radiation tend towards a state of *thermodynamic equilibrium*. This equilibrium state is achieved when sufficient interactions take place between the material particles ('collisions') and between the photons and mass par... | {
"Header 1": "**3.1 Local thermodynamic equilibrium**",
"token_count": 768,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The equation of state (EOS) describes the microscopic properties of stellar matter, for given density ρ, temperature *T* and composition *X<sup>i</sup>* . It is usually expressed as the relation between the pressure and these quantities:
$$P = P(\rho, T, X_i) \tag{3.1}$$
Using the laws of thermodynamics, and a simi... | {
"Header 1": "**3.2 The equation of state**",
"token_count": 571,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We shall derive the equation of state for a perfect gas from the principles of statistical mechanics. This provides a description of the ions, the electrons, as well as the photons in the deep stellar interior.
<sup>1</sup>N.B. note the difference between (local) *thermodynamic equilibrium* (*T*gas(*r*) = *T*rad(*r*)... | {
"Header 1": "**3.3 Equation of state for a gas of free particles**",
"token_count": 1076,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In general, the particle energies and velocities are related to their momenta according to special relativity:
$$\epsilon^2 = p^2 c^2 + m^2 c^4, \qquad \epsilon_p = \epsilon - mc^2 \tag{3.9}$$
and
$$v_p = \frac{\partial \epsilon}{\partial p} = \frac{pc^2}{\epsilon}.$$
(3.10)
We can obtain generally valid relati... | {
"Header 1": "**3.3 Equation of state for a gas of free particles**",
"Header 2": "3.3.1 Relation between pressure and internal energy",
"token_count": 962,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The ideal gas relation was derived for identical particles of mass *m*. It should be obvious that for a mixture of free particles of different species, it holds for the partial pressures of each of the constituents of the gas separately. In particular, it holds for both the ions and the electrons, as long as quantum-me... | {
"Header 1": "**3.3.3 Mixture of ideal gases, and the mean molecular weight**",
"token_count": 1140,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
According to quantum mechanics, the accuracy with which a particle's location and momentum can be known simultaneously is limited by Heisenberg's uncertainty principle, i.e. ∆*x*∆*p* ≥ *h*. In three dimensions, this means that if a particle is located within a volume element ∆*V* then its localization within three-dime... | {
"Header 1": "**3.3.4 Quantum-mechanical description of the gas**",
"token_count": 1826,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In the limit that $T \to 0$ , all available momentum states are occupied up to a maximum value, while all higher states are empty, as illustrated in the right panel of Fig. 3.2. This is known as *complete degeneracy*, and the maximum momentum is called the *Fermi momentum* $p_{\rm F}$ . Then we have
$$n_{\rm e}(p) ... | {
"Header 1": "**3.3.4 Quantum-mechanical description of the gas**",
"Header 2": "Complete electron degeneracy",
"token_count": 1885,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
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