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The luminosity L of a star can be derived from its radiative flux received at Earth and its distance using the relation
$$L = 4\pi d^2 f_{\text{bol}} = 4\pi R^2 \sigma T_{\text{eff}}^4 \text{ (in erg s}^{-1}$$
(2.3a)
or
$$L/L_{\odot} = (R/R_{\odot})^2 (T_{\text{eff}}/5777 \text{ K})^4 \text{ with } L_{\odot} = 3.... | {
"Header 1": "Observations of Stellar Parameters",
"Header 3": "2.3 The Luminosity of Stars",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The outer layer of a star from which radiation escapes is called the photosphere. It has a density and temperature structure with both quantities decreasing outward. The temperature structure can be derived from a detailed study of the spectrum
of the star. For understanding stellar evolution, we are mainly intereste... | {
"Header 1": "2.4 Magnitude, Color, and Temperature",
"token_count": 1966,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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Observations show that the luminosity of a star is related to its mass. This correlation is particularly strong for stars that are fusing H in their center. These are the main-sequence stars. The empirical relation, based on double lined spectroscopic main-sequence binaries, is shown in Figure 2.3.
The figure shows a... | {
"Header 1": "2.5 The Mass-Luminosity Relation",
"token_count": 360,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The Danish astronomer Ejnar Hertzsprung (1873–1967) in Copenhagen and the US astronomer Henry Norris Russell (1877–1959) at Princeton University independently discovered in 1905 and 1913 that stars occupy specific regions in the color– magnitude diagram (CMD). Since the colors are related to Teff and the magnitudes of ... | {
"Header 1": "2.6 The Hertzsprung–Russell Diagram and the Color–Magnitude Diagram",
"token_count": 1113,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The different regions in the HRD or CMD that are occupied by stars have been given different names to distinguish stellar types. Later, we will show that each of these types refers to a specific evolutionary phase with its corresponding internal structure. Figure 2.7 shows the regions and their names.
The stars are c... | {
"Header 1": "2.7 Nomenclature of Regions in the HRD and CMD",
"token_count": 410,
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1. The observed brightness of a star is expressed in magnitudes and its color is expressed by the difference between magnitudes measured in different filters.
- Brighter objects have smaller magnitudes. If the distance of a star is known, the apparent magnitude can be converted into an absolute magnitude.
- 2. The ab... | {
"Header 1": "2.8 Summary",
"token_count": 1011,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The equation of hydrostatic equilibrium follows from the equation of motion with the condition that the net force is zero. The equation of motion of one cm<sup>3</sup> gas with density $\rho$ in the shell at distance r and with a thickness dr can be written as
$$\rho \frac{d^2r}{dt^2} = -\rho \frac{GM_r}{r^2} - \fr... | {
"Header 1": "3.2 Hydrostatic Equilibrium",
"token_count": 1621,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The **virial theorem (VT)** links the total potential energy of a star in HE to the total internal (kinetic) energy of the star. We will see later that this helps with understanding why stars expand or contract during certain evolutionary phases. The word "virial" is derived from the Latin word of "vis" meaning "energy... | {
"Header 1": "3.3 The Virial Theorem: A Consequence of HE",
"token_count": 1377,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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- 1. The central pressure of a star in HE is on the order of Pc ∼ GM<sup>2</sup> /R<sup>4</sup> .
- 2. If the pressure in a star in HE is dominated by gas pressure, then the central temperature is on the order of *T m k GM R <sup>c</sup>* ∼ ( / )( / ) *μ* <sup>H</sup> .
- 3. A star in hydrostatic equilibrium must obey ... | {
"Header 1": "3.3 The Virial Theorem: A Consequence of HE",
"Header 3": "3.4 Summary",
"token_count": 931,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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In this chapter, we describe the relation between the major properties of the gas inside stars, i.e., density ρ, temperature T, and pressure P. This relation is called the equation of state (EoS). As the conditions in stars cover a wide range in characteristics, the EoS varies between different regimes in the (T, P, ρ)... | {
"Header 1": "Gas Physics of Stars",
"token_count": 256,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The mean particle mass depends on the composition and the degree of ionization of the gas. It is convenient to express the mean particle mass in atomic mass units (AMU). An AMU is so close to a proton mass that we will use m<sup>H</sup> for simplicity (see Table [1.1\)](#page-22-1).
The atomic mass Aj is also express... | {
"Header 1": "Gas Physics of Stars",
"Header 3": "4.1 Mean Particle Mass",
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The pressure exerted by particles or photons can be derived by considering a small volume, a box of $1 \times 1 \times 1$ cm<sup>3</sup>, filled with particles of velocity v. When a particle hits the side of a box at an impact angle $\theta$ and bounces off, its momentum $2mv \cos \theta$ is transferred to the wa... | {
"Header 1": "4.2 A General Expression for the Pressure",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The uncertainty principle is named after the German theoretical physicist Werner Heisenberg (1901–1976), who published it in 1927 at the age of 26.
#### 2. Pauli exclusion principle:
No two identical particles (same quantum state) can exist at the same time and place (i.e., in the same phase-space volume $h^3$ ). ... | {
"Header 1": "4.2 A General Expression for the Pressure",
"token_count": 1939,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
We can derive the electron density $n_e^{\rm crit}$ at which the gas goes from CD to RD by requiring
$$P_e(CD) = P_e(RD) \to K_1(\rho/\mu_e)^{5/3} = K_2(\rho/\mu_e)^{4/3} \to \to (\rho/\mu_e)_{crit} = (K_2/K_1)^3 = 1.91 \times 10^6 \text{ g cm}^{-3}.$$
(4.24)
Similarly, the boundary between the ideal gas law and ... | {
"Header 1": "4.2 A General Expression for the Pressure",
"token_count": 1992,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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#### 4.8.1 Proof That a Fully Convective Adiabatic Star Is a Polytrope
The energy transport in some types of stars or some stellar layers is not via the transport of radiation, but by convection: hot rising cells release their energy at the top of the convection zones and cool descending cells gain energy in deeper... | {
"Header 1": "4.2 A General Expression for the Pressure",
"token_count": 1259,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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1. The equation of state (EoS) describes the relation between P, T, and $\rho$ . At low densities this relation is given by the ideal gas law: $P \sim \rho T$ .
- 2. Radiation pressure is proportional to $T^4$ . It only plays an important role in the cores of massive stars and supernovae.
- 3. At densities in exce... | {
"Header 1": "4.2 A General Expression for the Pressure",
"Header 3": "4.9 Summary",
"token_count": 948,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The **absorption coefficient per gram** at frequency $\nu$ , $\kappa_{\nu}$ (in cm<sup>2</sup> g<sup>-1</sup>), is defined as the cross section for absorption or scattering of photons of frequency $\nu$ if these photons pass through a gram of gas. We can also define the **absorption coefficient per cm<sup>3</sup>*... | {
"Header 1": "5.1 The Rosseland-mean Opacity",
"token_count": 898,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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Free electrons scatter photons. The absorption cross section for electron scattering is $\sigma_e$ (in cm<sup>2</sup> g<sup>-1</sup>). Deep inside stars, the gas is fully ionized and electron scattering is the dominant opacity with
$$\sigma_e = \sigma_{\rm T} N_e, \tag{5.5}$$
where $\sigma_T = 6.65 \times 10^{-2... | {
"Header 1": "5.2 Electron Scattering: $\\sigma_e$",
"token_count": 849,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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If a photon with sufficient energy hits an atom that is not fully ionized it may kick out an electron. This photoionization results in the bound–free absorption of photons. The Rosseland-mean value of the bound–free absorption is calculated by summing all possible bound–free transitions of many ions. The result is know... | {
"Header 1": "5.2 Electron Scattering: $\\sigma_e$",
"Header 3": "5.4 Bound–Free Absorption: $\\kappa_{\\rm bf}$",
"token_count": 250,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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If the gas inside a star is not fully ionized, it can absorb photons of many possible wavelengths, resulting in electron transitions from one bound state to another. The calculation of this Rosseland-mean opacity is very difficult due to the numerous possible transitions of different ionization stages for the various e... | {
"Header 1": "5.5 Bound–Bound Absorption: $\\kappa_{\\rm bb}$",
"token_count": 1721,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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In 1926, the British astrophysicist Sir Arthur Eddington (1882–1944) derived an expression for the energy transport in stars by means of radiation. This is one of the fundamental equations for understanding the structure and evolution of stars. We start by giving this expression and then it will be derived intuitively ... | {
"Header 1": "6.1 Eddington's Equation for Radiative Equilibrium",
"token_count": 2041,
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} |
For any star in HE, the inward force due to gravity should be larger than the outward force by radiation pressure, because HE requires
$$\left| \frac{dP_{\text{rad}}}{dr} \right| + \left| \frac{dP_{\text{gas}}}{dr} \right| = \frac{GM(r)}{r^2} \rho_r \tag{6.8}$$
and so
$$\left| \frac{dP_{\text{rad}}}{dr} \right| <... | {
"Header 1": "6.3 The Eddington Limit: The Maximum Luminosity and the Maximum Mass",
"token_count": 1013,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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1. If the energy inside a star is transported outward by radiation, a specific T-gradient is required that is described by the Eddington Equation (6.1).
- 2. For a star in which the energy transport is by radiation, the combination of the hydrostatic and radiative equilibrium results in a relation between luminosity ... | {
"Header 1": "6.3 The Eddington Limit: The Maximum Luminosity and the Maximum Mass",
"Header 3": "6.4 Summary",
"token_count": 740,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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In most stars, energy is transported outward by radiation. In those layers where the transport of energy by radiation is not efficient enough, convection will take over. Hot bubbles rise from deeper layers, deliver their heat in higher layers, and descend again as cooler bubbles. A similar process occurs in the Earth's... | {
"Header 1": "Convective Energy Transport",
"token_count": 1668,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
We have derived the Schwarzschild criterion for convection by considering the rise of bubbles in a medium of constant chemical composition. We now consider the case of a chemically stratified star with a mean particle mass $\mu$ decreasing radially outward.
**Q** (7.3) Why would the mean particle mass decrease with... | {
"Header 1": "7.2 Convection in a Layer with a µ-gradient: Ledoux Criterion",
"token_count": 958,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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A rising convective cell will dissolve into its surroundings (i.e., it will lose its identity as a distinct cell), as the temperature of the gas inside the cell gradually adjusts to the temperature of its surroundings by the loss of radiation or heat at its boundary. The distance a hot cell rises or a cold cell descend... | {
"Header 1": "7.3 The Mixing Length: How Far Does a Convective Cell Rise before It Dissolves?",
"token_count": 1759,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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Above we derived an estimate for the product $\Delta T \cdot v_c$ . Thus, if we can estimate $v_c$ , we know $\Delta T$ . We can estimate the convection velocity by considering a toy model in which we treat a convective cell as a balloon. When a helium-filled balloon is released, it accelerates upward but quickly re... | {
"Header 1": "7.5 The Convective Velocity",
"token_count": 818,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
To get a "feeling" for the properties of a convective flow, let us estimate the values of $v_c$ and $\Delta T$ at a characteristic radius of $r = 0.5R_{\odot}$ (although the Sun is not convective at that radius).
$$g(0.5R_{\odot}) \simeq 10^{5} \text{ cm s}^{-2}$$
$$\ell_{p} \approx 0.1R_{\odot} = 7 \times 10... | {
"Header 1": "7.6 Typical Values of Convective Velocity and the Timescale",
"token_count": 742,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The mean T-gradient of the surroundings in a convection zone must be steeper than $(dT/dr)_{ad}$ of the convective cells; otherwise, the convection would stop.
**Q** (7.6) Explain why this is required.
If (dT/dr) describes the mean temperature gradient in a convection zone, then the Schwarzschild criterion requir... | {
"Header 1": "7.7 The Super-adiabatic Temperature Gradient in Convection Zones",
"token_count": 1972,
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We have seen that the timescale for the rise and descent of convection cells is much shorter than the nuclear timescale. This results in fast mixing throughout convection zones; as a result, a convective region is chemically homogeneous. Figure [7.8](#page-80-1) shows this in a schematic way for stars with convective e... | {
"Header 1": "7.10 Chemical Mixing by Convection and Its Consequences",
"token_count": 408,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
1. Convection occurs in the layers where energy transport by radiation would require the T-gradient to be steeper than the adiabatic one: ∣ ∣ >∣ ∣ *dT dr dT dr* / / rad ad. This is the Schwarzschild criterion. In those layers, energy is transported outward by hotter rising cells and cooler descending cells. Convective ... | {
"Header 1": "7.11 Summary",
"token_count": 1070,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
During nuclear fusion two particles (i and j) react, which results in one or two other particles (k and l). The particles involved have a charge Z and a mass A.
So the reaction is i + j → k + l, with nuclear charge conservation Zi + Zj = Zk + Zl, and baryon number conservation: Ai + Aj = Ak + Al.
The reaction rate,... | {
"Header 1": "8.1 Reaction Rates and Energy Production",
"token_count": 1270,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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Ions have a positive charge, so they will repulse one another by electric Coulomb forces. To enable fusion, the particles must overcome this Coulomb barrier. Figure [8.2](#page-85-2) shows the potential due to the repulsive Coulomb force that increases toward smaller distance atr > rn. The maximum potential is Ec = E(r... | {
"Header 1": "8.1 Reaction Rates and Energy Production",
"Header 3": "8.2 Thermonuclear Reaction Rates and the Gamow Peak",
"token_count": 2048,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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Along the way $^{15}\text{N}$ nuclei are formed, from which the ON-cycle can deviate. $^{17}\text{O}$ is the starting point of ON-cycle II and $^{18}\text{O}$ is the starting point of ON-cycle III. All of these cycles produce a $^4\text{He}$ nucleus and return to an earlier reaction in the chain.
The CN-cycle... | {
"Header 1": "8.1 Reaction Rates and Energy Production",
"Header 3": "8.2 Thermonuclear Reaction Rates and the Gamow Peak",
"token_count": 2049,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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(8.17a)
The terms in parentheses (in, out) indicate what is added to the original nucleus and what comes out in addition to the resulting nucleus. At $T < 3 \times 10^7$ K, the abundance of $^{22}$ Ne decreases and the abundance of $^{23}$ Na increases.
The MgAl cycle proceeds as
$$^{24}\text{Mg} \quad (\math... | {
"Header 1": "8.1 Reaction Rates and Energy Production",
"Header 3": "8.2 Thermonuclear Reaction Rates and the Gamow Peak",
"token_count": 324,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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At $T > 10^8$ K, <sup>4</sup>He fuses into <sup>12</sup>C via the reactions shown in Table 8.2.
This is called the triple- $\alpha$ process. The mass defect of the net reaction is $\Delta m/m = 0.00065$ .
The first reaction is an equilibrium reaction that results in a very small fraction of Be ions. The mean li... | {
"Header 1": "8.5 He $\\rightarrow$ C Fusion: The Triple- $\\alpha$ Process",
"token_count": 2037,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The table also lists the products and the amount of net energy produced per nucleon, corrected for the energy loss by neutrinos, in
| | | | 1 | | | | ... | {
"Header 1": "8.5 He $\\rightarrow$ C Fusion: The Triple- $\\alpha$ Process",
"token_count": 2028,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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We also see that nuclei in this diagram that have an unstable isotope on their left and are *not* shielded by a stable isotope in the diagonal direction can only be formed by the r-process (e.g., $^{148}$ Nd and $^{150}$ Nd). Some nuclei can be formed by both processes (e.g., $^{151}$ Sm and $^{152}$ Sm).
#### s-... | {
"Header 1": "8.5 He $\\rightarrow$ C Fusion: The Triple- $\\alpha$ Process",
"token_count": 422,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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We have seen in Table 8.4 that each reaction requires a minimum temperature to be ignited. The central temperature of a star can rise if the star, or rather its core, contracts. We have derived before (from HE and the ideal gas law, in Section 3.2) that we can estimate the central temperature of a star as
$$T_c \sime... | {
"Header 1": "8.10 The Minimum Core Mass for Igniting Fusion Reactions",
"token_count": 1127,
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At the end of each nuclear fusion phase, the core of a star contracts to release potential energy, compensating for the outward transport of energy. When the core density increases, its temperature also increases. The core contraction stops when it becomes degenerate. From then on, $\rho_c$ hardly increases. As there... | {
"Header 1": "8.10 The Minimum Core Mass for Igniting Fusion Reactions",
"Header 3": "8.11 Fusion Phases of Stars in the $(\\rho_c, T_c)$ Plane",
"token_count": 2033,
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Suppose that the gas pressure in part of the star or in the whole star suddenly vanishes. This is the most drastic possible disturbance of hydrostatic equilibrium (HE), apart from an explosion. How much time would it take for the star to restore HE? If the pressure vanishes in all layers simultaneously the star would g... | {
"Header 1": "9.1 The Dynamical Timescale",
"token_count": 1987,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The estimates given above show that the dynamical timescale, the convection timescale, the Kelvin–Helmholtz timescale, and the nuclear timescales are very different, and that
$$\tau_{\rm dyn} \ll \tau_{\rm conv} \ll \tau_{\rm KH} \ll \tau_{\rm nucl}$$
(9.6)
The values for the Sun are 1 hr ≪ weeks ≪ 3.10<sup>7</sup>... | {
"Header 1": "9.5 Comparison of Timescales",
"token_count": 259,
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The stellar models that we describe here are simple and valid for most evolutionary phases. They are based on the following assumptions.
- 1. The star is spherically symmetric, which implies that
- the physical quantities vary only in the radial direction: P(r), ρ(r), T(r), etc., and
- the effects of rotation and mag... | {
"Header 1": "10.1 Assumptions for Computing Stellar Evolution",
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| | in <i>r</i><br>Euler coordinates | in $m = M(r)$<br>Lagrange coordinates | |
|----------------------------------|---------------------------------... | {
"Header 1": "10.1 Assumptions for Computing Stellar Evolution",
"Header 3": "**10.2** The Equations of Stellar Structure",
"token_count": 1130,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The equations of stellar structure in Lagrangian coordinates consist of four differential equations for r(m), P(m), L(m), and T(m). Solving these equations for a star of mass M requires four boundary conditions:
$$r(m=0) = 0, (10.8a)$$
$$L(m=0) = 0, (10.8b)$$
$$P(m = M) = 0, (10.8c)$$
$$T(m = M) = T_{\text{eff}... | {
"Header 1": "10.3 Boundary Conditions",
"token_count": 1789,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The predicted stellar evolution of a star is based on the computation of a series of models in hydrostatic equilibrium and thermal equilibrium with varying chemical structure. This has two interesting consequences.
1. The structure of a star in HE and TE depends only on the chemical profile as a function of mass. It ... | {
"Header 1": "10.5 Principles of Stellar Evolution Calculations",
"token_count": 445,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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- 1. The structure of a star with a given mass and chemical structure in HE and TE is described by a set of four differential equations. These can be expressed in Lagrangian coordinates, with m = M(r) as the running parameter.
- 2. There are four boundary conditions: r = 0 and L = 0 in the center, where m = 0, and P = ... | {
"Header 1": "10.6 Summary",
"token_count": 239,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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In Section 4.8, we showed that under certain physical conditions the equation of state (EoS) can be written as $P \sim \rho^{\gamma}$ . If a star has such a polytropic EoS, its density and pressure structure follows from the hydrostatic equilibrium (HE) equation. This solution of the HE equation does not provide infor... | {
"Header 1": "Polytropic Stars",
"token_count": 263,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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If the EoS can be expressed as a polytrope $P = K\rho^{\gamma}$ , the pressure is independent of temperature and thus analytical solutions to the HE equation exist for certain values of $\gamma$ .
In this case, the HE can be combined with the mass continuity equation
$$\frac{r^2}{\rho} \frac{dP}{dr} = -Gm \to \fr... | {
"Header 1": "11.1 The Structure of Polytropic Stars: $P = K\\rho^{\\gamma}$",
"token_count": 1442,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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The solution of the Lane–Emden equation yields $\theta(\xi)$ with $\rho = \rho_c \theta^n$ and $r = \xi/\alpha$ . The inner boundary condition at the center of the star is $\theta = 1$ at $\xi = 0$ .
The radius of the star is defined by $\rho = 0$ , so $\theta(\xi_1) = 0$ , and $\xi_1$ is the value of the... | {
"Header 1": "11.2 Stellar Parameters of Polytropic Models",
"token_count": 887,
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Polytropic stars have a mass-radius relation that depends on n or $\gamma$ . This follows from Equation (11.10a), which shows that $M \sim \alpha^3 \rho_c$ . Equation (11.8) shows that $R \sim \alpha$ and Equation (11.5) shows that $\rho_c^{(1-n)/n} \sim \alpha^2$ . Combining these equations shows that the **mass-... | {
"Header 1": "11.3 The Mass-Radius Relation of Polytropic Stars",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
A polytrope model with n = 3 (i.e., $P = K\rho^{4/3}$ ) and with a fixed constant K has the curious property that it can exist for only one particular mass. This follows from the substitution of Equation (11.5) for $\alpha$ into Equation (11.10) for M. This is the case for extreme relativistic degenerate stars, wher... | {
"Header 1": "11.3.4 Polytropes with $\\gamma = 4/3$ and n = 3 and Fixed K: Relativistic Degenerate Stars",
"token_count": 200,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
We have shown in Section 4.8 that stars in which the ratio $1-\beta=P_{\rm rad}/P$ is constant behave as polytropes with $P \sim aT^4/3$ and $P \sim \rho$ T, so $P \sim \rho^{4/3}$ (i.e., $\gamma=4/3$ and n=3). Sir Arthur Eddington assumed that $P_{\rm rad}/P=1-\beta$ is constant in stars and derived the po... | {
"Header 1": "11.3.5 Polytropes with $\\gamma = 4/3$ and n = 3 and Variable K: Eddington's Standard Model",
"token_count": 1795,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. The density, pressure, and mass structure of stars with a polytropic EoS of the type $P = K\rho^{\gamma}$ for any value of $\gamma$ or $n = 1/(\gamma 1)$ is fixed. The mass–radius relation depends on n as listed in Table 11.2.
- 2. For degenerate stars, the value of K is fixed by atomic constants and so the ... | {
"Header 1": "11.4 Summary",
"token_count": 740,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
In this chapter, we discuss the process of star formation. Stars form out of cold molecular interstellar clouds when the inward gravitational force is stronger than the outward force due to gas pressure and magnetic pressure. This leads to an initial collapse on a free-fall timescale and the subsequent fragmentation of... | {
"Header 1": "Star Formation",
"token_count": 1759,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
A cloud of a given mass will contract and start forming stars if it exceeds the Jeans mass, which depends on density as $M_{\rm J} \sim n^{-1/2}$ (Equation 12.6). The cloud contraction and star formation are triggered by compression due to shocks, which increases the density. The three major triggering mechanisms are... | {
"Header 1": "**12.3** The Collapse of Molecular Clouds",
"token_count": 903,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The cooling mechanisms prevent the adiabatic heating of a cloud when it collapses. The potential energy gained by the contractions is immediately emitted, so the collapse proceeds approximately isothermally. The increasing density of isothermal clouds implies that the Jeans mass decreases so that substructures of the c... | {
"Header 1": "**12.3** The Collapse of Molecular Clouds",
"Header 3": "12.4 Fragmentation of Molecular Clouds",
"token_count": 2044,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Let us assume for simplicity that $A \approx 2$ .
In fact, the collapsing cloud loses most of the gained energy in the form of far-IR photons. Only a fraction f < 1 of $\Delta E_{\rm pot}$ is used for dissociation and ionization. If the clump was contracting quasi-hydrostatically, then f would have been 0.5 accord... | {
"Header 1": "**12.3** The Collapse of Molecular Clouds",
"Header 3": "12.4 Fragmentation of Molecular Clouds",
"token_count": 816,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
When the dissociation and ionization of the star is complete, the star is in hydrostatic equilibrium. The star does not have nuclear fusion yet, but it has a temperature gradient so it radiates. The star must therefore contract to cover this energy loss.
We have argued that at this phase the star is fully convective.... | {
"Header 1": "12.7 The Contraction of a Convective Protostar: The Descent along the Hayashi Track",
"token_count": 1442,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
At the end of the Hayashi phase of a star, when $\bar{T} \sim 10^6$ K, the convection gradually stops and the protostar goes into radiative equilibrium. The protostar in radiative equilibrium has not yet started nuclear fusion, so it will keep contracting to cover the loss of energy by radiation. Because it is in rad... | {
"Header 1": "12.8 The Contraction of a Radiative Pre-main-sequence Star: From the Hayashi Track to the Main Sequence",
"token_count": 1937,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Sometime later, at a core temperature of about $2.5 \times 10^6$ K (i.e., when the star is near the $10^6$ yr isochrones), Li is destroyed by the reaction chain
$$^{6}\text{Li}(p, \gamma) \,^{7}\text{Be}(e^{-}, \nu) \,^{7}\text{Li}(p, \gamma) \,^{8}\text{Be} \rightarrow 2^{4}\text{He} + \text{energy}.$$
(12.19) ... | {
"Header 1": "12.8 The Contraction of a Radiative Pre-main-sequence Star: From the Hayashi Track to the Main Sequence",
"token_count": 396,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Hydrogen fusion requires a minimum $T_c$ of about $6 \times 10^6$ K. We have shown in Section 8.10 that this requires some minimum mass. This minimum mass is $0.08 M_{\odot}$ . If the mass is lower, the core becomes degenerate and the contraction stops before the required $T_c$ is reached. Stars with $M < 0.08 ... | {
"Header 1": "12.11 Stars That Do Not Reach H-fusion: Brown Dwarfs with $M < 0.08~M_{\\odot}$",
"token_count": 2038,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
At densities n<1 cm<sup>-3</sup>, the H<sub>2</sub> fraction is very small, viz. $<10^{-5}$ , and the cooling of contracting clouds is very inefficient. Therefore, T rises almost adiabatically as the cloud contracts so $T \sim n^{2/3}$ (Equation (4.33)) until it reaches a virial temperature of about 5000 K. At n>1 c... | {
"Header 1": "12.11 Stars That Do Not Reach H-fusion: Brown Dwarfs with $M < 0.08~M_{\\odot}$",
"token_count": 598,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
1. We summarize the important phases in the process of star formation, with the characteristic timescales and characteristic stellar parameters for a star of $1M_{\odot}$ .
Collapse of cloud = free fall phase $\tau_{\rm ff} \sim 10^5 \ {\rm yr}$ : Cooling by molecules and dust IR radiation. $\bar{T}$ is low (abou... | {
"Header 1": "**12.14 Summary**",
"token_count": 1222,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
When a PMS star arrives on the main sequence at the start of the core H-fusion, it is still chemically homogeneous, apart from minor changes due to D-fusion. The location of homogeneous stars in the HRD is called the **zero-age main sequence**, **ZAMS**. The properties of stars on the ZAMS can be understood on the basi... | {
"Header 1": "13.1 The Zero-age Main Sequence (ZAMS): Homology Relations",
"token_count": 1851,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
\tag{13.9}$$
Comparing the above expression for L with $L \sim \mu^4 M^3/\kappa$ (Equation (13.3)) for stars in radiative equilibrium, we get
$$\frac{\mu^{\nu} M^{2+\nu}}{R^{3+\nu}} \sim \frac{\mu^4 M^3}{\kappa}.$$
(13.10)
This yields an expression for R
$$R \sim \mu^{\frac{\nu-4}{3+\nu}} \frac{1}{\kappa^{3+\... | {
"Header 1": "13.1 The Zero-age Main Sequence (ZAMS): Homology Relations",
"token_count": 1220,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The dependence on metallicity Z. The results above show that for massive ZAMS stars Teff is rather insensitive to metallicity (Z), but for intermediate mass ZAMS stars a lower Z implies a higher Teff and a larger L.
The dependence on helium abundance Y. The dependence of Teff and L on X and μ implies that Teff and L ... | {
"Header 1": "13.2 The Influence of Abundances on the ZAMS",
"token_count": 992,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The main-sequence phase is the longest phase in the evolution of a star. During this phase the star changes slightly because its internal chemical composition is changing. The changes in the chemical composition of the star depend on the presence or absence of convection in the core. A star without a convective core cr... | {
"Header 1": "13.3 Evolution during the Main-sequence Phase",
"token_count": 2047,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
In Section 7.9, we showed that a high value of $L/4\pi r^2$ was one of the reasons for convection.
MS stars with $M > 1.2 M_{\odot}$ have a convective core. MS stars with $M < 1.2 M_{\odot}$ have a radiative core.
N.B. The convective region in the center of stars with $M \gtrsim 2M_{\odot}$ is larger than t... | {
"Header 1": "13.3 Evolution during the Main-sequence Phase",
"token_count": 2032,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. At the start of the core H-fusion phase, stars occupy a narrow strip in the HRD called the ZAMS. The location of the ZAMS depends on the initial chemical composition and can be explained by homology relations.
- 2. The ZAMS of stars with lower metal abundance or higher He abundance is shifted to the left and upwar... | {
"Header 1": "13.6 Summary",
"token_count": 280,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 13.1 Explain in physical terms the difference in main-sequence lifetime between stars of Z = 0.014 and Z = 0.002 plotted in Figure [13.9.](#page-153-1)
- 13.2 We will show later (Section 25) that differential rotation in a fast rotating star may result in severe mixing during the main-sequence phase.
Calculate the ... | {
"Header 1": "13.6 Summary",
"Header 3": "Exercises",
"token_count": 487,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
At the end of the MS phase, a star has a core with mass $M_c$ and radius $R_c$ that no longer produces nuclear energy. This implies $L(r) \approx 0$ for $r < R_c$ . If $L(r < R_c) \approx 0$ , then $dT/dr \approx 0$ in the core; otherwise, there would be transport of radiation. This shows that a core without ... | {
"Header 1": "14.1 Isothermal Cores: The Schönberg-Chandrasekhar Limit",
"token_count": 2029,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- When the core contracts, the envelope expands.
- When the core expands, the envelope contracts.
There are two ways to explain this: using the virial theorem argument or using the pressure argument. In both arguments, the thermostatic behavior of a fusion shell plays a key role (see Section 13.3.1).
#### The vir... | {
"Header 1": "14.1 Isothermal Cores: The Schönberg-Chandrasekhar Limit",
"token_count": 1903,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
So, the density $\rho_1$ at $\tau = 1$ is $\rho_1 = 1/\mathcal{H}\bar{\kappa} \sim GM/TR^2\bar{\kappa}$ and the gas pressure at $R_1 \equiv R$ ( $\tau = 1$ ) due to the photosphere is
$$\boxed{P_1^{\text{phot}} \sim \rho_1 T_1 \sim \frac{GM}{R^2} \frac{1}{\overline{\kappa}}} \quad \text{with} \quad \rho_1 \si... | {
"Header 1": "14.1 Isothermal Cores: The Schönberg-Chandrasekhar Limit",
"token_count": 2028,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. After the MS phase, when H-fusion stops in the core but continues in a shell, the core is nearly isothermal. Isothermal cores can only remain in hydrostatic equilibrium if the ratio Mc/M < 0.10, where Mc is the core mass. This is the Schönberg–Chandrasekhar limit. If M<sup>c</sup> > 0.10 M, the core must contract.... | {
"Header 1": "14.4 Summary",
"token_count": 423,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The evolution of stars is strongly affected by mass loss, so we must first discuss the physics of stellar winds and the resulting mass-loss rates before continuing the description of post-MS stellar evolution. Stars may lose mass due to a stellar wind. The mass-loss rates are particularly high in luminous hot stars, wh... | {
"Header 1": "Stellar Winds and Mass Loss",
"token_count": 330,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The theory and observations of stellar winds are described in the book "Introduction to Stellar Winds" (Lamers & Cassinelli, 1999). There are four basic types of stellar winds.
- a. **Line-driven winds** of hot stars are driven by radiation pressure on highly ionized atoms of C, N, O, and Fe-group elements. This mech... | {
"Header 1": "15.1 Types of Winds",
"token_count": 2018,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
As a result, hot luminous stars may have mass-loss rates slightly higher than the value of $\dot{M}_{\rm max}$ derived above.
O (15.1) Sketch the path of a scattered photon that transfers more momentum to the wind than $h\nu/c$ .
#### 15.2.2 Observed Wind Velocities and Mass-loss Rates of Hot Stars
The observe... | {
"Header 1": "15.1 Types of Winds",
"token_count": 788,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Cool stars can have dust in their outer layers. The dust-to-gas ratio is typically ∼10−<sup>2</sup> by mass, because dust cannot be formed from H and He but only from metals. Dust is an efficient absorber of radiation, so the absorption of stellar photons by dust grains produces a transfer of momentum (i.e., radiation ... | {
"Header 1": "15.3 Dust-driven Winds of Cool Stars",
"token_count": 2028,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
If there is no
Table 15.2 The Condensation Radius of Dust in Cool Star Winds.
| Dust type $T_{\rm cond}$ | Silicate 1100 K | Amorphous Carbon 1500 K | Graphite 1500 K |
|--------------------------|-----------------|-------------------------|-----------------|
| $T_{ m eff}$ | $r_c/R$ | $r_c/R$ ... | {
"Header 1": "15.3 Dust-driven Winds of Cool Stars",
"token_count": 2041,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Winds of cool supergiants and AGB stars have $v_{\infty} \approx 10~{\rm km~s^{-1}} \ll v_{\rm esc}$ . This shows that the kinetic energy of the winds is much smaller than the potential energy. The Reimers relation thus implies that for these stars a *fixed fraction of the stellar luminosity* is used to provide the *p... | {
"Header 1": "15.3 Dust-driven Winds of Cool Stars",
"token_count": 2049,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Color–magnitude diagrams and Hertzsprung–Russell diagrams of old star clusters invariably show a significant fraction of stars on the red giant branch. These stars are brighter and cooler than the main-sequence stars in the same cluster. This indicates that stars increase their radii after the phase of core H-fusion. R... | {
"Header 1": "Shell H-fusion in Low- and Intermediate-mass Stars: Red Giants",
"token_count": 255,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
When H is exhausted in the core, the core contracts. The layers around it also contract and their temperature increases to above T > 107 K, high enough for H-shell fusion. We have shown in Section [13.4](#page-152-1) that H-shell fusion starts differently in stars with M ≳ 1.2M<sup>ʘ</sup> and stars with M ≲ 1.2M<sup>ʘ... | {
"Header 1": "Shell H-fusion in Low- and Intermediate-mass Stars: Red Giants",
"Header 3": "16.1 The Start of the H-shell Fusion",
"token_count": 2021,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The post-MS evolution of stars with an initial mass of 2≲ Mi≲ 8M<sup>ʘ</sup> is very similar to that of lower-mass stars. The tracks have the same general structure as in Figure [16.1.](#page-179-0) The stars go through the subgiant phase and the red giant phase when the star is largely convective and evolves upward on... | {
"Header 1": "16.3 The H-shell Fusion Phase of Intermediate-mass Stars of 2–8 M<sup>ʘ</sup>",
"token_count": 560,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Stars with Mi ≲ 2M<sup>ʘ</sup> have a degenerate He core and a very extended envelope when they are on the RGB. This implies that the pressure on top of the H-fusion shell is small. The pressure at the bottom of the H shell is mainly due to the shell itself and the gravity from the core. This implies that the efficienc... | {
"Header 1": "16.4 The Mcore–L Relation for Red Giants",
"token_count": 265,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
We have seen that fully convective stars are on the Hayashi line. Red giants are almost fully convective stars because their core contains only a small fraction of the stellar mass. We have shown in Section [14.3](#page-159-1) that the location of the Hayashi line in the HRD depends strongly on the opacity in the photo... | {
"Header 1": "16.5 Metallicity Dependence of the Red Giant Branch",
"token_count": 767,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Red giants lose mass by means of a stellar wind. These stars are located in the HRD on the red giant branch (RGB). The winds of subgiants and low-luminosity RGB stars are possibly driven by gas pressure in a chromosphere combined with dissipation of magnetic Alfvén waves that originate in the convection zone (Lamers & ... | {
"Header 1": "16.6 Mass Loss during the Red Giant Phase",
"token_count": 318,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
High accuracy photometric observations obtained with the Hubble Space Telescope (HST) of stars in old globular clusters have shown that the CMDs of most old globular clusters do not agree with the prediction of a single population of co-eval stars formed from a cloud with a unique abundance pattern. The main sequence i... | {
"Header 1": "16.7 Multiple Populations in Globular Clusters",
"token_count": 1414,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. At the end of the core H-fusion phase the core contracts and H-fusion starts in a shell around the He core.
- 2. In stars of M ≲ 2Mʘ, the core mass is higher than the Schönberg– Chandrasekhar limit, so the core contracts. Due to the mirror action of the shell, the envelope expands by deepening the convective layer... | {
"Header 1": "16.8 Summary",
"token_count": 945,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
When the mass of the He core reaches a value of about $0.5M_{\odot}$ , slightly dependent on the total stellar mass, the contracting core reaches the ignition temperature of He-fusion at $T_c \sim 10^8$ K. Because of the core mass-luminosity relation of red giants, this happens at the tip of the RGB at $L \sim 10^3... | {
"Header 1": "17.1 The Ignition of Helium Fusion in Low-mass Stars",
"token_count": 2043,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Figure 16.1 shows that the increase and decrease of the radius is produced by a decrease and later an increase of the mass and depth of the convective envelope during phase G-H.
The slight increase in luminosity during core He-fusion is due to the same $\mu$ -effect, $L \sim \overline{\mu}^4$ , that is responsible ... | {
"Header 1": "17.1 The Ignition of Helium Fusion in Low-mass Stars",
"token_count": 873,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Because the He-fusion phase is much shorter than the H-fusion phase, all HB stars in a cluster at any time come from a small range in initial masses. This implies that they
Figure 17.3. Blue lines show the predicted evolutionary tracks of HB stars with the same helium core mass of 0.49M... | {
"Header 1": "17.1 The Ignition of Helium Fusion in Low-mass Stars",
"Header 3": "17.4 The Observed HB of Globular Clusters",
"token_count": 1935,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
At the end of the HB phase, when He is exhausted in the core, the resulting CO core is without an energy source so it will contract. Because the star still has a H-fusing shell with mirror action, the core contraction results in envelope expansion and the star moves to the right in the HRD. Because the expanding envelo... | {
"Header 1": "Double Shell Fusion: Asymptotic Giant Branch Stars",
"Header 3": "18.1 The Start of the AGB Phase",
"token_count": 580,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The He-fusion shell is directly on top of the degenerate CO core. This implies that the pressure in the shell and its energy production are mainly set by the mass of the core. This is analogous to the case of the H-fusion shell on top of the degenerate He core in a low-mass RGB star (Section 16.4).
The $M_c$ –L rela... | {
"Header 1": "Double Shell Fusion: Asymptotic Giant Branch Stars",
"Header 3": "18.2 The M<sub>core</sub>-L Relation of AGB Stars",
"token_count": 2046,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
While the He-shell fusion is going on and the ISZ is expanding, the pressure in the He-shell, mainly produced by the overlaying ISZ, reduces. Eventually, the pressure on the He-shell is so low that the energy production in the He-shell decreases and the layers on top of it contract again. The increase in T and $\rho$ ... | {
"Header 1": "Double Shell Fusion: Asymptotic Giant Branch Stars",
"Header 3": "18.2 The M<sub>core</sub>-L Relation of AGB Stars",
"token_count": 1310,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The occurrence of thermal pulses with the alternate switching between He shell and Hshell fusionleads to efficientmixing and gradual changesin the atmospheric abundances. This process occurs in two steps, as shown in Figure [18.6](#page-205-0).
- 1. During the thermal pulse, when the ISZ is convective for a short tim... | {
"Header 1": "18.5 The Third Dredge-up",
"token_count": 683,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
It is useful to summarize the dredge-up phases of stars of 0.8 ≲ Mi ≲ 8Mʘ. Figure [18.7](#page-206-2) shows, schematically, the position in the HRD where they occur.
Figure 18.6. Internal evolution of an AGB star, explaining the dredge-up during two thermal pulses. The thick and thin re... | {
"Header 1": "18.6 Summary of the Dredge-up Phases",
"token_count": 705,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Throughout the AGB phase, the luminosity of the star increases. Because the luminosity during the AGB phase is set by the core mass, while the growth of the core mass is set by the luminosity, there is a simple way to estimate the speed with which a star ascends the AGB-branch.
Let us assume that the luminosity of an... | {
"Header 1": "18.6 Summary of the Dredge-up Phases",
"Header 3": "18.7 The Evolution Speed during the AGB Phase",
"token_count": 1963,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Assuming $T_{\rm eff} = 4000$ K for Galactic AGB stars, we find
$$\dot{M}_{\rm w} \simeq 1.2 \times 10^{-10} \, (L/L_{\odot})^{1.05} = 3.4 \times 10^{-7} \, e^{t/1.33 \text{Myr}} \, M_{\odot}/\text{yr}.$$
(18.8)
Integration of this equation shows that the envelope has lost $2.4M_{\odot}$ after 2.3Myr. At that t... | {
"Header 1": "18.6 Summary of the Dredge-up Phases",
"Header 3": "18.7 The Evolution Speed during the AGB Phase",
"token_count": 2030,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
After the AGB phase, when a star has lost almost all of its very extended convective envelope, the remaining envelope mass shrinks. The luminosity remains constant because it is still produced by shell fusion on top of a degenerate core. The shrinking of the envelope implies a decreasing radius, so the star moves horiz... | {
"Header 1": "Post-AGB Evolution and Planetary Nebulae",
"token_count": 1769,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
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