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As a star, or its core, contracts the density may become so high that the electrons become degenerate and exert a (much) higher pressure than they would if they behaved classically. Since in the limit of strong degeneracy the pressure no longer depends on the temperature, this degeneracy pressure can hold the star up a... | {
"Header 1": "Importance of electron degeneracy in stars",
"token_count": 355,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Photons can be treated as quantum-mechanical particles that carry momentum and therefore exert pressure when they interact with matter. In particular photons are *bosons* with $g_s = 2$ (two polarization states), so they can be described by the Bose-Einstein statistics, eq. (3.27). The number of photons is not conser... | {
"Header 1": "3.3.6 Radiation pressure",
"token_count": 1719,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
It is often important to consider processes that occur on such a short (e.g. hydrodynamical) timescale that there is no heat exchange with the environment; such processes are *adiabatic*. To derive the properties of stellar interiors under adiabatic conditions we need several thermodynamic derivatives. We therefore sta... | {
"Header 1": "3.4 Adiabatic processes",
"token_count": 724,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The specific heats at constant volume $c_V$ and at constant pressure $c_P$ for a unit mass of gas follow from eq. (3.47):
$$c_V = \left(\frac{\mathrm{d}q}{\mathrm{d}T}\right)_v = \left(\frac{\partial u}{\partial T}\right)_v,\tag{3.51}$$
$$c_P = \left(\frac{\mathrm{d}q}{\mathrm{d}T}\right)_P = \left(\frac{\parti... | {
"Header 1": "3.4.1 Specific heats",
"token_count": 665,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The thermodynamic response of a system to adiabatic changes is measured by the so-called *adiabatic derivatives*. Two of these have special importance for stellar structure:
• The *adiabatic exponent*<sup>2</sup> <sup>γ</sup>ad measures the response of the pressure to adiabatic compression or expansion, i.e. to a cha... | {
"Header 1": "**3.4.2 Adiabatic derivatives**",
"token_count": 1857,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have so far implicitly assumed complete ionization of the gas, i.e. that it consists of bare atomic nuclei and free electrons. This is a good approximation in hot stellar interiors, where *T* > 10<sup>6</sup> K so that typical energies *kT* are much larger than the energy needed to ionize an atom, i.e. to knock off ... | {
"Header 1": "**3.5 Ionization**",
"token_count": 402,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As an example, we consider the simple case where the gas consists only of hydrogen. Then there are just three types of particle, electrons and neutral and ionized hydrogen, with *u*<sup>H</sup> = *u*<sup>0</sup> = 2 and *u*H<sup>+</sup> = *u*<sup>1</sup> = 1. We write their number densities as *n*<sup>+</sup> and *n*<s... | {
"Header 1": "**3.5.1 Ionization of hydrogen**",
"token_count": 1691,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In a mixture of gases the situation becomes more complicated because many, partly ionized species have to be considered, the densities of which all depend on each other (see e.g. K&W Chapter 14.4-14.5). However the basic physics remains the same as considered above for the simple case of pure hydrogen. The effect on th... | {
"Header 1": "3.5.2 Ionization of a mixture of gases",
"token_count": 526,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have so far ignored the effect of electrostatic or Coulomb interactions between the ions and electrons in the gas. Is this a reasonable approximation, i.e. are the interaction energies indeed small compared to the kinetic energies, as we have assumed in Sect. 3.3?
The average distance between gas particles (with m... | {
"Header 1": "3.6.1 Coulomb interactions and crystallization",
"token_count": 1008,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
If $\Gamma_C \gg 1$ the thermal motions of the ions are overwhelmed by the Coulomb interactions. In this situation the ions will tend to settle down into a conglomerate with a lower energy, in other words they will form a crystalline lattice. Detailed estimates indicate that this transition takes place at a critical ... | {
"Header 1": "Crystallization",
"token_count": 293,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
A very different process can take place at very high temperatures and relatively low densities. A photon may turn into an electron-positron pair if its energy hv exceeds the rest-mass energy of the pair, $hv > 2m_{\rm e}c^2$ . This must take place during the interaction with a nucleus, since otherwise momentum and ene... | {
"Header 1": "3.6.2 Pair production",
"token_count": 395,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The contents of this chapter are also covered by Chapter 7 of Maeder and by Chapters 13 to 16 of Kippenhahn & Weigert. However, a more elegant derivation of the equation of state, which is also more consistent with the way it is derived in these lecture notes, is given in Chapter 3 of Hansen, Kawaler & Trimble. Explici... | {
"Header 1": "3.6.2 Pair production",
"Header 2": "Suggestions for further reading",
"token_count": 1066,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
(a) Use the first law of thermodynamics to show that, for an ideal gas in an adiabatic process,
$$P \propto \rho^{\gamma_{\rm ad}}$$
(3.74)
and give a value for the adiabatic exponent γad.
(b) Use the ideal gas law in combination with eq. (3.74) to show that
$$\nabla_{\rm ad} = \left(\frac{\mathrm{d} \ln T}{\ma... | {
"Header 1": "3.6.2 Pair production",
"Header 2": "**3.5 Adiabatic derivatives**",
"token_count": 389,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In this Appendix we derive some of the thermodynamic relations that were given without proof in Chapter 3.
The first law of thermodynamics states that the heat added to a mass element of gas is the sum of the change in its internal energy and the work done by the mass element. Taking the element to be of unit mass, w... | {
"Header 1": "3.A Appendix: Themodynamic relations",
"token_count": 1337,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
It is useful to be able to write the change in heat content of a unit mass in terms of the changes in the state variables. Eq. (3.77) already shows how d*q* is written in terms of *T* and ρ, i.e.
$$dq = T ds = c_V dT - \chi_T \frac{P}{\rho^2} d\rho, \tag{3.85}$$
making use of (3.79) and (3.80). It is often useful t... | {
"Header 1": "3.A Appendix: Themodynamic relations",
"Header 2": "**Expressions for d***q*",
"token_count": 404,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Eq. (3.88) makes it easy to derive an expression for the adiabatic temperature gradient (3.57),
$$\nabla_{\rm ad} \equiv \left(\frac{\mathrm{d}\ln T}{\mathrm{d}\ln P}\right)_{\rm ad}.\tag{3.89}$$
An adiabatic change in *T* and *P* means the changes take place at constant *s*, or with d*q* = 0. Hence (3.88) shows th... | {
"Header 1": "**Adiabatic derivatives**",
"token_count": 768,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As mentioned in Sec. 2.2, the equation of hydrostatic equilibrium can be solved if the pressure is a known function of the density, *P* = *P*(ρ). In this situation the mechanical structure of the star is completely determined. A special case of such a relation between *P* and ρ is the *polytropic relation*,
$$P = K\r... | {
"Header 1": "**Polytropic stellar models**",
"token_count": 286,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
If the equation of state can be written in polytropic form, the equations for mass continuity (d*m*/d*r*, eq. 2.3) and for hydrostatic equilibrium (d*P*/d*r*, eq. 2.12) can be combined with eq. (4.1) to give a second-order differential equation for the density:
$$\frac{1}{\rho r^2} \frac{\mathrm{d}}{\mathrm{d}r} \lef... | {
"Header 1": "**4.1 Polytropes and the Lane-Emden equation**",
"token_count": 1584,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Once the solution *w*(*z*) of the Lane-Emden equation is found, eq. (4.5) fixes the relative density distribution of the model, which is thus uniquely determined by the polytropic index *n*. Given the solution for a certain *n*, the physical properties of a polytropic stellar model, such as its mass and radius, are the... | {
"Header 1": "**4.1.1 Physical properties of the solutions**",
"token_count": 1308,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Eq. (4.15) expresses a relation between the constant *K* in eq. (4.1) and the mass and radius of a polytropic model. This relation can be interpreted in two very different ways:
- The constant *K* may be given in terms of physical constants. This is the case, for example, for a star dominated by the pressure of degen... | {
"Header 1": "**4.2 Application to stars**",
"token_count": 234,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Stars that are so compact and dense that their interior pressure is dominated by degenerate electrons are known observationally as *white dwarfs*. They are the remnants of stellar cores in which hydrogen has been completely converted into helium and, in most cases, also helium has been fused into carbon and oxygen. Sin... | {
"Header 1": "**4.2.1 White dwarfs and the Chandrasekhar mass**",
"token_count": 643,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As an example of a situation where K is not fixed by physical constants but is essentially a free parameter, we consider a star in which the pressure is given by a mixture of ideal gas pressure and radiation pressure, eq. (3.45). In particular we make the assumption that the ratio $\beta$ of gas pressure to total pre... | {
"Header 1": "4.2.2 Eddington's standard model",
"token_count": 638,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
#### **4.1 The Lane-Emden equation**
- (a) Derive eq. (4.2) from the stellar structure equations for mass continuity and hydrostatic equilibrium. (Hint: multiply the hydrostatic equation by *r* 2 /ρ and take the derivative with respect to *r*).
- (b) What determines the second boundary condition of eq. (4.4), i.e., w... | {
"Header 1": "**Exercises**",
"token_count": 369,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
To understand some of the properties of white dwarfs (WDs) we start by considering the equation of state for a degenerate, non-relativistic electron gas.
- (a) What is the value of *K* for such a star? Remember to consider an appropriate value of the mean molecular weight per free electron µ*e*.
- (b) Derive how the ... | {
"Header 1": "**4.3 White dwarfs**",
"token_count": 479,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The energy that a star radiates from its surface is generally replenished from sources or reservoirs located in its hot central region. This represents an outward energy flux at every layer in the star, and it requires an effective means of transporting energy through the stellar material. This transfer of energy is po... | {
"Header 1": "**Energy transport in stellar interiors**",
"token_count": 306,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In Chapter 2 we considered the global energy budget of a star, regulated by the virial theorem. We have still to take into account the conservation of energy on a local scale in the stellar interior. To do this we turn to the first law of thermodynamics (Sect. 3.4), which states that the internal energy of a system can... | {
"Header 1": "**5.1 Local energy conservation**",
"token_count": 1428,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have seen that most stars are in a long-lived state of thermal equilibrium, in which energy generation in the stellar centre exactly balances the radiative loss from the surface. What would happen if the nuclear energy source in the centre is suddenly quenched? The answer is: very little, at least initially. Photons... | {
"Header 1": "**5.2 Energy transport by radiation and conduction**",
"token_count": 464,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Fick's law of diffusion states that, when there is a gradient $\nabla n$ in the density of particles of a certain type, the diffusive flux J – i.e. the net flux of such particles per unit area per second – is given by
$$J = -D \nabla n$$
with $D = \frac{1}{3} \bar{\nu} \ell$ . (5.9)
Here D is the diffusion coeff... | {
"Header 1": "**5.2 Energy transport by radiation and conduction**",
"Header 2": "5.2.1 Heat diffusion by random motions",
"token_count": 740,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
For photons, we can take $\bar{v} = c$ and $U = aT^4$ . Hence the specific heat (per unit volume) is $C_V = dU/dT = 4aT^3$ . The photon mean free path can be obtained from the equation of radiative transfer, which states that the intensity $I_v$ of a radiation beam (in a medium without emission) is diminished ove... | {
"Header 1": "5.2.2 Radiative diffusion of energy",
"token_count": 1043,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The radiative diffusion equations derived above are independent of frequency $\nu$ , since the flux F is integrated over all frequencies. However, in general the opacity coefficient $\kappa_{\nu}$ depends on frequency, such that the $\kappa$ appearing in eq. (5.16) or (5.17) must represent a proper average over fr... | {
"Header 1": "5.2.3 The Rosseland mean opacity",
"token_count": 912,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Collisions between the gas particles (ions and electrons) can also transport heat. Under normal (ideal gas) conditions, however, the collisional conductivity is very much smaller than the radiative conductivity. The collisional cross sections are typically $10^{-18} - 10^{-20} \, \mathrm{cm}^2$ at the temperatures in... | {
"Header 1": "5.2.4 Conductive transport of energy",
"token_count": 693,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
An electromagnetic wave that passes an electron causes it to oscillate and radiate in other directions, like a classical dipole. This scattering of the incoming wave is equivalent to the effect of absorption, and can be described by the Thomson cross-section of an electron
$$\sigma_{\rm e} = \frac{8\pi}{3} \left(\fra... | {
"Header 1": "**5.3.1 Sources of opacity**",
"Header 2": "**Electron scattering**",
"token_count": 1325,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Bound-free absorption is the absorption of a photon by a bound electron whereby the photon energy exceeds the ionization energy $\chi$ of the ion or atom. Computing the opacity due to this process requires carefully taking into account the atomic physics of all the ions and atoms present in the mixture, and is thus v... | {
"Header 1": "**5.3.1 Sources of opacity**",
"Header 2": "Bound-free and bound-bound absorption",
"token_count": 1225,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In general, κ = κ(ρ, *T*, *Xi*) is a complicated function of density, temperature and composition. While certain approximations can be made, as in the examples shown above, these are usually too simplified and inaccurate to apply in detailed stellar models. An additional complication is that the Rosseland mean opacity ... | {
"Header 1": "**5.3.2 A detailed view of stellar opacities**",
"token_count": 907,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have seen that radiative transport of energy inside a star requires a temperature gradient dT/dr, the magnitude of which is given by eq. (5.16). Since $P_{\rm rad} = \frac{1}{3}aT^4$ , this means there is also a gradient in the radiation pressure:
$$\frac{\mathrm{d}P_{\mathrm{rad}}}{\mathrm{d}r} = -\frac{4}{3}aT^... | {
"Header 1": "5.4 The Eddington luminosity",
"token_count": 1147,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
For radiative diffusion to transport energy outwards, a certain temperature gradient is needed, given by eq. (5.16) or eq. (5.17). The larger the luminosity that has to be carried, the larger the temperature gradient required. There is, however, an upper limit to the temperature gradient inside a star – if this limit i... | {
"Header 1": "**5.5 Convection**",
"token_count": 218,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
So far we have assumed strict spherical symmetry in our description of stellar interiors, i.e. assuming all variables are constant on concentric spheres. In reality there will be small fluctuations, arising for example from the thermal motions of the gas particles. If these small perturbations do not grow they can safe... | {
"Header 1": "**5.5.1 Criteria for stability against convection**",
"token_count": 1474,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The stability criterion (5.44) is not of much practical use, because it involves computation of a density gradient which is not part of the stellar structure equations. We would rather have a criterion for the temperature gradient, because this also appears in the equation for radiative transport. We can rewrite eq. (5... | {
"Header 1": "**The Schwarzschild and Ledoux criteria**",
"token_count": 1230,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
According to the Schwarzschild criterion, we can expect convection to occur if
$$\nabla_{\text{rad}} = \frac{3}{16\pi a c G} \frac{P}{T^4} \frac{\kappa l}{m} > \nabla_{\text{ad}}.$$
(5.52)
This requires one of following:
- A large value of κ, that is, convection occurs in opaque regions of a star. Examples are th... | {
"Header 1": "Occurrence of convection",
"token_count": 1345,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Within the framework of MLT we can calculate the convective energy flux, and the corresponding temperature gradient required to carry this flux, as follows. After rising over a radial distance $\ell_m$ the temperature difference between the gas element (e) and its surroundings (s) is
$$\Delta T = T_{\rm e} - T_{\rm... | {
"Header 1": "The convective energy flux",
"token_count": 1967,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Besides being an efficient means of transporting energy, convection is also a very efficient *mixing* mechanism. We can see this by considering the average velocity of convective cells, eq. (5.56), and taking $\ell_{\rm m} \approx H_P$ and $\sqrt{gH_P} \approx v_{\rm s}$ , so that
$$v_{\rm c} \approx v_{\rm s} \sq... | {
"Header 1": "The convective energy flux",
"Header 2": "5.5.3 Convective mixing",
"token_count": 1047,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
To determine the extent of a region that is mixed by convection, we need to look more closely at what happens at the boundary of a convective zone. According to the Schwarzschild criterion derived in Sec. 5.5.1, in a chemically homogeneous layer this boundary is located at the surface where $\nabla_{\rm rad} = \nabla_... | {
"Header 1": "**5.5.4** Convective overshooting",
"token_count": 507,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The most important way to transport energy form the interior of the star to the surface is by radiation, i.e. photons traveling from the center to the surface.
(a) How long does it typically take for a photon to travel from the center of the Sun to the surface? [Hint: estimate the mean free path of a photon in the ce... | {
"Header 1": "Suggestions for further reading",
"Header 3": "5.1 Radiation transport",
"token_count": 1080,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
- (a) Why does convection lead to a net heat flux upwards, even though there is no net mass flux (upwards and downwards bubbles carry equal amounts of mass)?
- (b) Explain the Schwarzschild criterion
$$\left(\frac{d \ln T}{d \ln P}\right)_{\text{rad}} > \left(\frac{d \ln T}{d \ln P}\right)_{\text{ad}}$$
in simple p... | {
"Header 1": "Suggestions for further reading",
"Header 2": "5.4 Conceptual questions: convection",
"token_count": 215,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
- (a) Low-mass stars, like the Sun, have convective envelopes. The fraction of the mass that is convective increases with decreasing mass. A $0.1~M_{\odot}$ star is completely convective. Can you qualitatively explain why?
- (b) In contrast higher-mass stars have radiative envelopes and convective cores, for reasons ... | {
"Header 1": "5.5 Applying Schwarzschild's criterion",
"token_count": 658,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
For a star in thermal equilibrium, an internal energy source is required to balance the radiative energy loss from the surface. This energy source is provided by *nuclear reactions* that take place in the deep interior, where the temperature and density are sufficiently high. In ordinary stars, where the idealgas law h... | {
"Header 1": "**Nuclear processes in stars**",
"token_count": 231,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Consider a reaction whereby a nucleus *X* reacts with a particle *a*, producing a nucleus *Y* and a particle *b*. This can be denoted as
$$X + a \rightarrow Y + b$$
or $X(a,b)Y$ . (6.1)
The particle *a* is generally another nucleus, while the particle *b* could also be a nucleus, a γ-photon or perhaps another kin... | {
"Header 1": "**6.1 Basic nuclear properties**",
"token_count": 324,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The masses of atomic nuclei are not exactly equal to the sum of the masses of the individual nucleons (protons and neutrons), because the nucleons are bound together by the strong nuclear force. If *m<sup>i</sup>* denotes the mass of a nucleus *i*, then the *binding energy* of the nucleus can be defined as
$$E_{B,i} ... | {
"Header 1": "**6.1.1 Nuclear energy production**",
"token_count": 1614,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Consider again a reaction of the type *X*(*a*, *b*)*Y*. Let us first suppose that particles *X* are bombarded by particles *a* with a particular velocity υ. The rate at which they react then depends on the *crosssection*, i.e. the effective surface area of the particle *X* for interacting with particle *a*. The cross-s... | {
"Header 1": "**6.2 Thermonuclear reaction rates**",
"token_count": 1007,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The cross-section $\sigma$ appearing in the reaction rate equation (6.8) is a measure of the probability that a nuclear reaction occurs, given the number densities of the reacting nuclei. While the energy gain from a reaction can be simply calculated from the mass deficits of the nuclei, the cross-section is much mor... | {
"Header 1": "**6.2.1** Nuclear cross-sections",
"token_count": 1451,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
A typical thermonuclear reaction proceeds as follows. After penetrating the Coulomb barrier, the two nuclei can from an unstable, excited *compound nucleus* which after a short time decays into the product particles, e.g.
$$X + a \to C^* \to Y + b .$$
Although the lifetime of the compound nucleus *C* ∗ is very shor... | {
"Header 1": "**Nuclear structure e**ff**ects on the cross-section**",
"token_count": 1309,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Since λ <sup>2</sup> <sup>∝</sup> <sup>1</sup>/*<sup>E</sup>* and *<sup>P</sup>*(*E*) <sup>∝</sup> exp(−*b E*−1/<sup>2</sup> ), one usually writes
$$\sigma(E) = S(E) \frac{\exp(-b E^{-1/2})}{E} \qquad (6.19)$$
This equation defines the 'astrophysical *S* -factor' *S* (*E*), which contains all remaining effects, i.e... | {
"Header 1": "**The astrophysical cross-section factor**",
"token_count": 1148,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The value of the Gamow peak energy $E_0$ can be found by taking df/dE = 0, which gives
$$E_0 = (\frac{1}{2}bkT)^{2/3} = 5.665 (Z_i^2 Z_i^2 A T_7^2)^{1/3} \text{ keV}.$$
(6.23)
To obtain the last equality we have substituted b as given by eq. (6.15) and we use the notation $T_n = T/(10^n \, \text{K})$ , while A i... | {
"Header 1": "**The astrophysical cross-section factor**",
"Header 2": "Properties of the Gamow peak",
"token_count": 1986,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We found that the repulsive Coulomb force between nuclei plays a crucial role in determining the rate of a thermonuclear reaction. In our derivation of the cross section we have ignored the influence of the surrounding free electrons, which provide overall charge neutrality in the gas. In a dense medium, the attractive... | {
"Header 1": "**6.2.3** Electron screening",
"token_count": 692,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Having obtained an expression for the cross-section factor $\langle \sigma v \rangle$ , the reaction rate $r_{ij}$ follows from eq. (6.8). We can then easily obtain the energy generation rate. Each reaction releases an amount of energy $Q_{ij}$ according to eq. (6.4), so that $Q_{ij} r_{ij}$ is the energy genera... | {
"Header 1": "**6.3** Energy generation rates and composition changes",
"token_count": 596,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The reaction rates also determine the rate at which the composition changes. The rate of change in the number density *n<sup>i</sup>* of nuclei of type *i* owing to reactions with nuclei of type *j* is
$$\left(\frac{\mathrm{d}n_i}{\mathrm{d}t}\right)_j = -(1+\delta_{ij})\,r_{ij} = -n_i n_j \,\langle \sigma \nu \rangl... | {
"Header 1": "**6.3** Energy generation rates and composition changes",
"Header 2": "**Composition changes**",
"token_count": 1018,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In principle, many different nuclear reactions can occur simultaneously in a stellar interior. If one is interested in following the detailed isotopic abundances produced by all these reactions, or if structural changes occur on a very short timescale, a large network of reactions has to be calculated (as implied by eq... | {
"Header 1": "**6.4 The main nuclear burning cycles**",
"token_count": 236,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The net result of hydrogen burning is the fusion of four <sup>1</sup>H nuclei into a <sup>4</sup>He nucleus,
$$4^{1}H \rightarrow {}^{4}He + 2e^{+} + 2\nu$$
. (6.44)
You may verify using Sec. 6.1.1 that the total energy release is 26.734 MeV. In order to create a <sup>4</sup>He nucleus two protons have to be conver... | {
"Header 1": "**6.4.1 Hydrogen burning**",
"token_count": 1796,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
If some C, N, and O is already present in the gas out of which a star forms, and if the temperature is sufficiently high, hydrogen fusion can take place via the so-called *CNO cycle*. This is a cyclical sequence of reactions that typically starts with a proton capture by a <sup>12</sup>C nucleus, as follows:
$$\begin... | {
"Header 1": "**6.4.1 Hydrogen burning**",
"Header 2": "The CNO cycle",
"token_count": 1888,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Helium burning consists of the fusion of $^4$ He into a mixture of $^{12}$ C and $^{16}$ O, which takes place at temperatures $T \gtrsim 10^8$ K. Such high temperatures are needed because (1) the Coulomb barrier for He fusion is higher than that of the H-burning reactions considered above, and (2) fusion of $^4$ ... | {
"Header 1": "**6.4.1 Hydrogen burning**",
"Header 2": "6.4.2 Helium burning",
"token_count": 1198,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In the mixture of mainly $^{12}$ C and $^{16}$ O that is left after helium burning, further fusion reactions can occur if the temperature rises sufficiently. In order of increasing temperature, the nuclear burning cycles that may follow are the following.
**Carbon burning** When the temperature exceeds $T_8 \gtrsi... | {
"Header 1": "**6.4.1 Hydrogen burning**",
"Header 2": "6.4.3 Carbon burning and beyond",
"token_count": 1740,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Neutrinos play a special role because their cross-section for interaction with normal matter is extremely small. The neutrinos that are released as a by-product of nuclear reactions have typical energies in the MeV range, and at such energies the interaction cross-section is $\sigma_{\nu} \sim 10^{-44} \, \mathrm{cm}^... | {
"Header 1": "6.5 Neutrino emission",
"token_count": 1491,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
#### **6.1 Conceptual questions: Gamow peak**
N.B. Discuss your answers to this question with your fellow students or with the assistant.
In the lecture (see eq. 6.22) you saw that the reaction rate is proportional to
$$\langle \sigma \upsilon \rangle = \left(\frac{8}{m\pi}\right)^{1/2} \frac{S(E_0)}{(kT)^{3/2}} ... | {
"Header 1": "**Exercises**",
"token_count": 370,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Estimate the relative abundances of the nuclei CN-equilibrium if their lifetimes against proton capture at *<sup>T</sup>* <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup><sup>7</sup> K are: <sup>τ</sup>p( N) = 30 yr, τp( C) = 1600 yr, τp( C) = 6600 yr and τp( N) <sup>=</sup> <sup>6</sup> <sup>×</sup> <sup>10</sup><... | {
"Header 1": "**6.3 Relative abundances for CN equilibrium**",
"token_count": 654,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In the previous chapters we have reviewed the most important physical processes taking place in stellar interiors, and we derived the differential equations that determine the structure and evolution of a star. By putting these ingredients together we can construct models of spherically symmetric stars. Because the com... | {
"Header 1": "Stellar models and stellar stability",
"token_count": 221,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Let us collect and summarize the differential equations for stellar structure and evolution that we have derived in the previous chapters, regarding m as the spatial variable, i.e. eqs. (2.6), (2.11), (5.4), (5.17) and (6.41):
$$\frac{\partial r}{\partial m} = \frac{1}{4\pi r^2 \rho} \tag{7.1}$$
$$\frac{\partial P}... | {
"Header 1": "7.1 The differential equations of stellar evolution",
"token_count": 1790,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
At the centre (*m* = 0), both the density and the energy generation rate must remain finite. Therefore, both *r* and *l* must vanish in the centre:
$$m = 0$$
: $r = 0$ and $l = 0$ . (7.6)
However, nothing is known a priori about the central values of *P* and *T*. Therefore the remaining two boundary conditions m... | {
"Header 1": "**7.2.1 Central boundary conditions**",
"token_count": 271,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
At the surface (*m* = *M*, or *r* = *R*), the boundary conditions are generally much more complicated than at the centre. One may treat the surface boundary conditions at different levels of sophistication.
- The simplest option is to take *T* = 0 and *P* = 0 at the surface (the 'zero' boundary conditions). However, ... | {
"Header 1": "**7.2.2 Surface boundary conditions**",
"token_count": 578,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
It is instructive to look at the effect of the surface boundary conditions on the solution for the structure of the outer envelope of a star. Assuming complete (dynamical and thermal) equilibrium, the envelope contains only a small fraction of the mass and no energy sources. In that case l = L and $m \approx M$ . It i... | {
"Header 1": "**7.2.2 Surface boundary conditions**",
"Header 2": "7.2.3 Effect of surface boundary conditions on stellar structure",
"token_count": 1571,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
For a star in both hydrostatic and thermal equilibrium, the four partial differential equations for stellar structure (eqs. 7.1–7.4) reduce to ordinary, time-independent differential equations. We can further simplify the situation somewhat, by ignoring possible neutrino losses ( $\epsilon_{\nu}$ ) which are only impor... | {
"Header 1": "7.3 Equilibrium stellar models",
"token_count": 838,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Solving the stellar structure equations almost always requires heavy numerical calculations, such as are applied in detailed stellar evolution codes. However, there is often a kind of similarity between the numerical solutions for different stars. These can be approximated by simple analytical scaling relations known a... | {
"Header 1": "**7.4 Homology relations**",
"token_count": 1729,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In order to obtain simple homology relations from the other structure equations, we must make additional assumptions. We start by analysing eq. (7.15).
• First, let us assume the *ideal gas* equation of state,
$$P = \frac{\mathcal{R}}{\mu} \rho T.$$
Let us further assume that in each star the *composition is homo... | {
"Header 1": "**7.4.1 Homology for radiative stars composed of ideal gas**",
"token_count": 1078,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
For stars that are in thermal equilibrium we can make use of the last structure equation (7.14) to derive further homology relations for the radius as a function of mass. We then have to assume a specific form for the energy generation rate, say
$$\epsilon_{\text{nuc}} = \epsilon_0 \, \rho T^{\nu} \tag{7.34}$$
so t... | {
"Header 1": "**7.4.2 Main sequence homology**",
"token_count": 672,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have seen in Chapter 2 that, as a consequence of the virial theorem, a star without internal energy sources must contract under the influence of its own self-gravity. Suppose that this contraction takes place homologously. According to eq. (7.17) each mass shell inside the star then maintains the same relative radiu... | {
"Header 1": "**7.4.3 Homologous contraction**",
"token_count": 618,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The question of dynamical stability relates to the response of a certain part of a star to a perturbation of the balance of forces that act on it: in other words, a perturbation of hydrostatic equilibrium. We already treated the case of dynamical stability to *local* perturbations in Sec. 5.5.1, and saw that in this ca... | {
"Header 1": "7.5.1 Dynamical stability of stars",
"token_count": 782,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Stars dominated by an ideal gas or by non-relativistic degenerate electrons have $\gamma_{ad} = \frac{5}{3}$ and are therefore dynamically stable. However, we have seen that for relativistic particles $\gamma_{ad} \to \frac{4}{3}$ and stars dominated by such particles tend towards a neutrally stable state. A small ... | {
"Header 1": "Cases of dynamical instability",
"token_count": 375,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The question of thermal or *secular* stability, i.e. the stability of thermal equilibrium, is intimately linked to the virial theorem. In the case of an ideal gas the virial theorem (Sect. 2.3) tells us that the total energy of a star is
$$E_{\text{tot}} = -E_{\text{int}} = \frac{1}{2}E_{\text{gr}},\tag{7.44}$$
whi... | {
"Header 1": "7.5.2 Secular stability of stars",
"token_count": 2034,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
\tag{7.54}$$
The shell is thermally stable as long as expansion results in a drop in temperature, i.e. when
$$4\frac{d}{r} > \chi_{\rho} \tag{7.55}$$
since χ*<sup>T</sup>* > 0. Thus, for a sufficiently thin shell a thermal instability will develop. (In the case of an ideal gas, the condition 7.55 gives *d*/*r* > ... | {
"Header 1": "7.5.2 Secular stability of stars",
"token_count": 210,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The differential equations (7.1–7.5) describe, for a certain location in the star at mass coordinate *m*, the behaviour of and relations between radius coordinate *r*, the pressure *P*, the temperature *T*, the luminosity *l* and the mass fractions *X<sup>i</sup>* of the various elements *i*.
- (a) Which of these equ... | {
"Header 1": "**7.1 General understanding of the stellar evolution equations**",
"token_count": 298,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
- (a) Show that for a star in hydrostatic equilibrium $(dP/dm = -Gm/(4\pi r^4))$ the pressure scales with density as $P \propto \rho^{4/3}$ .
- (b) If $\gamma_{ad} < 4/3$ a star becomes dynamically unstable. Explain why.
- (c) In what type of stars $\gamma_{ad} \approx 4/3$ ?
- (d) What is the effect of partial i... | {
"Header 1": "**7.1 General understanding of the stellar evolution equations**",
"Header 2": "7.2 Dynamical Stability",
"token_count": 691,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Consider a star in hydrostatic equilibrium (HE), for which we can estimate how the central pressure scales with mass and radius from the homology relations (Sec. 7.4). For a star that expands or contracts homologously, we can apply eq. (7.26) to the central pressure and central density to yield
$$P_c = C \cdot GM^{2/... | {
"Header 1": "8.1.1 Hydrostatic equilibrium and the $P_c$ - $\\rho_c$ relation",
"token_count": 406,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
By considering the EOS we can also derive the evolution of the central temperature. This is obviously crucial for the evolutionary fate of a star because e.g. nuclear burning requires $T_c$ to reach certain (high) values. We start by considering lines of constant T, isotherms, in the $(P, \rho)$ plane.
We have en... | {
"Header 1": "8.1.2 The equation of state and evolution in the $P_c$ - $\\rho_c$ plane",
"token_count": 1416,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We now consider how the stellar centre evolves in the $T_c$ , $\rho_c$ diagram. First we divide the T, $\rho$ plane into regions where different processes dominate the EOS, see Sec. 3.3.7 and Fig. 3.4, reproduced in Fig. 8.2a.
For a slowly contracting star in hydrostatic equilibrium equation (8.1) implies that, ... | {
"Header 1": "**8.1.3** Evolution in the $T_c$ - $\\rho_c$ plane",
"token_count": 1365,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We found that stars with $M < M_{\rm Ch}$ reach a maximum temperature, the value of which increases with mass. This means that only gas spheres above a certain mass limit will reach temperatures sufficiently high for nuclear burning. The nuclear energy generation rate is a sensitive function of the temperature, which... | {
"Header 1": "8.2 Nuclear burning regions and limits to stellar masses",
"token_count": 931,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As a consequence of the virial theorem, a self-gravitating sphere composed of ideal gas in HE must contract and heat up as it radiates energy from the surface. The energy loss occurs at a rate
$$L = -\dot{E}_{\text{tot}} = \dot{E}_{\text{in}} = -\frac{1}{2}\dot{E}_{\text{gr}} \approx \frac{E_{\text{gr}}}{\tau_{\text{... | {
"Header 1": "8.2.1 Overall picture of stellar evolution and nuclear burning cycles",
"token_count": 2025,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Column (3) gives the total gravitational energy emitted per nucleon since the beginning, and column (5) the total nuclear energy emitted per nucleon since the beginning. Column (6) gives the minimum mass required to ignite a certain burning stage (column 4). The last two columns give the fraction of energy emitted as p... | {
"Header 1": "8.2.1 Overall picture of stellar evolution and nuclear burning cycles",
"token_count": 1174,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The process of star formation constitutes one of the main problems of modern astrophysics. Compared to our understanding of what happens *after* stars have formed out of the interstellar medium – that is, stellar evolution – star formation is a very ill-understood problem. No predictive theory of star formation exists,... | {
"Header 1": "**9.1 Star formation and pre-main sequence evolution**",
"token_count": 2030,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Somewhat later, further dynamical collapse phases follow when first H and then He are ionized at $\sim 10^4\,\mathrm{K}$ . When ionization of the protostar is complete it settles back into hydrostatic equilibrium at a much reduced radius (see below).
**Pre-main sequence phase** Finally, the accretion slows down and ... | {
"Header 1": "**9.1 Star formation and pre-main sequence evolution**",
"token_count": 811,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have seen in Sect. 7.2.3 that as the effective temperature of a star decreases the convective envelope gets deeper, occupying a larger and larger part of the mass. If $T_{\rm eff}$ is small enough stars can therefore become completely convective. In that case, as we derived in Sect. 5.5.2, energy transport is very... | {
"Header 1": "9.1.1 Fully convective stars: the Hayashi line",
"token_count": 2008,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Consider models in the neighbourhood of the Hayashi line in the H-R diagram for a star of mass M. These models cannot have $\nabla = \nabla_{\rm ad}$ throughout, because otherwise they would be on the Hayashi line. Defining $\bar{\nabla}$ as the average value of $d\log T/d\log P$ over the entire star, models on e... | {
"Header 1": "The forbidden region in the H-R diagram",
"token_count": 2041,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Stars on the zero-age main sequence are (nearly) homogeneous in composition and are in complete (hydrostatic and thermal) equilibrium. Detailed models of ZAMS stars can be computed by solving the four differential equations for stellar structure numerically. It is instructive to compare the properties of such models to... | {
"Header 1": "9.2 The zero-age main sequence",
"token_count": 1416,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We can estimate how the central temperature and central density scale with mass and composition for a ZAMS star from the homology relations for homogeneous, radiative stars in thermal equilibrium (Sec. 7.4.2, see eqs. 7.37 and 7.38 and Table 7.1). From these relations we may expect the central temperature to increase w... | {
"Header 1": "9.2 The zero-age main sequence",
"Header 2": "9.2.1 Central conditions",
"token_count": 708,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
An overview of the occurrence of convective regions on the ZAMS as a function of stellar mass is shown in Fig. 9.8. For any given mass M, a vertical line in this diagram shows which conditions are encountered as a function of depth, characterized by the fractional mass coordinate m/M. Gray shading indicates whether a p... | {
"Header 1": "9.2.2 Convective regions",
"token_count": 1144,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Fig. 9.9 shows the location of the ZAMS in the H-R diagram and various evolution tracks for different masses at Population I composition, covering the central hydrogen burning phase. Stars evolve away
from the ZAMS towards higher luminosities and larger radii. Low-mass stars ( $M \lesssim 1 M_{\odot}$ ) evolve toward... | {
"Header 1": "9.3 Evolution during central hydrogen burning",
"token_count": 828,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We can understand why rather massive stars ( $M \gtrsim 1.3 M_{\odot}$ ) expand during the MS by considering the pressure that the outer layers exert on the core:
$$P_{\text{env}} = \int_{m_{\star}}^{M} \frac{Gm}{4\pi r^4} \, \mathrm{d}m \tag{9.14}$$
Expansion of the envelope (increase in r of all mass shells) mean... | {
"Header 1": "9.3.1 Evolution of stars powered by the CNO cycle",
"token_count": 727,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In stars with $M \lesssim 1.3 \, M_{\odot}$ the central temperature is too low for the CNO cycle and the main energy-producing reactions are those of the pp chain. The lower temperature sensitivity $\epsilon_{\rm pp} \propto \rho \, T^4$ means that $T_c$ and $\rho_c$ increase more than was the case for the CNO ... | {
"Header 1": "9.3.2 Evolution of stars powered by the pp chain",
"token_count": 468,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The timescale τMS that a star spends on the main sequence is essentially the nuclear timescale for hydrogen burning, given by eq. (2.37). Another way of deriving essentially the same result is by realizing that, in the case of hydrogen burning, the rate of change of the hydrogen abundance *X* is related to the energy g... | {
"Header 1": "**9.3.3 The main sequence lifetime**",
"token_count": 1238,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As discussed in Sect. 5.5.4, the size of a convective region inside a star is expected to be larger than predicted by the Schwarzschild (or Ledoux) criterion because of convective *overshooting*. However, the extent $d_{ov}$ of the overshooting region is not known reliably from theory. In stellar evolution calculatio... | {
"Header 1": "**9.3.3 The main sequence lifetime**",
"Header 2": "9.3.4 Complications: convective overshooting and semi-convection",
"token_count": 814,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The process of star formation and pre-main sequence evolution is treated in much more detail in Chapters 18–20 of Maeder, while the properties and evolution on the main sequence are treated in Chapter 25. See also Kippenhahn & Weigert Chapters 22 and 26–30.
#### **Exercises**
#### 9.1 Kippenhahn diagram of the ZAMS... | {
"Header 1": "**9.3.3 The main sequence lifetime**",
"Header 2": "Suggestions for further reading",
"token_count": 553,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
After the main-sequence phase, stars are left with a hydrogen-exhausted core surrounded by a still hydrogen-rich envelope. To describe the evolution after the main sequence, it is useful to make a division based on the mass:
**low-mass stars** are those that develop a degenerate helium core after the main sequence, l... | {
"Header 1": "Post-main sequence evolution through helium burning",
"token_count": 426,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
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