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During central hydrogen burning on the main sequence, we have seen that stars are in thermal equilibrium ( $\tau_{\text{nuc}} \gg \tau_{\text{KH}}$ ) with the surface luminosity balanced by the nuclear power generated in the centre. After the main sequence a hydrogen-exhausted core is formed inside which nuclear energy... | {
"Header 1": "10.1 The Schönberg-Chandrasekhar limit",
"token_count": 1686,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In this section we discuss in some detail the evolution of stars during hydrogen-shell burning, until the onset of helium burning. Based on the above section, qualitative differences are to be expected between low-mass stars ( $M \lesssim 2 M_{\odot}$ ) on the one hand and intermediate- and high-mass stars ( $M \gtrsim... | {
"Header 1": "10.2 The hydrogen-shell burning phase",
"token_count": 373,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Fig. 10.2 shows the evolution track of a 5 *M*<sup>⊙</sup> star of quasi-solar composition (*X* = 0.7, *Z* = 0.02) in the H-R diagram, and Fig. 10.3 shows some of the interior details of the evolution of this star as a function of time from the end of central hydrogen burning. Point B in both figures corresponds to the... | {
"Header 1": "**10.2.1 Hydrogen-shell burning in intermediate-mass and massive stars**",
"token_count": 2061,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Schonberg-Chandrasekhar limit has become irrelevant. Therefore low-mas ¨ s stars can remain in HE and TE throughout hydrogen-shell burning and there is no Hertzsprung gap in the H-R diagram.
This can be seen in Fig. 10.5 which shows the internal evolution of a 1 *M*<sup>⊙</sup> star with quasi-solar composition in ... | {
"Header 1": "**10.2.1 Hydrogen-shell burning in intermediate-mass and massive stars**",
"token_count": 610,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The evolution of low-mass stars along the red giant branch is very similar and almost independent of the mass of the star. The reason for this similarity is that by the time the helium core has become degenerate, a very strong density contrast has developed between the core and the envelope. The envelope is so extended... | {
"Header 1": "10.2.3 The red giant branch in low-mass stars",
"token_count": 1088,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
When the convective envelope reaches its deepest extent at point D in Fig. 10.5, it has penetrated into layers that were processed by H-burning during the main sequence, and have been partly processed by the CN-cycle. Up to point D the surface He abundance increases and the H abundance decreases, but more noticeably th... | {
"Header 1": "**First dredge-up and the luminosity bump**",
"token_count": 239,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Another process that becomes important in low-mass red giants is *mass loss*. As the stellar luminosity and radius increase as a star evolves along the giant branch, the envelope becomes loosely bound and it is relatively easy for the large photon flux to remove mass from the stellar surface. The process driving mass l... | {
"Header 1": "**Mass loss on the red giant branch**",
"token_count": 347,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We again take the 5 $M_{\odot}$ star depicted in Figs. 10.2–10.3 as a typical example of an intermediate-mass star. The ignition of helium takes place at point E in these figures. Since the core is non-degenerate at this point ( $\rho_c \approx 10^4 \, \text{g/cm}^3$ , Fig. 10.1), nuclear burning is thermally stable ... | {
"Header 1": "10.3.1 Helium burning in intermediate-mass stars",
"token_count": 1731,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In low-mass stars (with $M \lesssim 2 \, M_{\odot}$ ) the helium burning phase differs from more massive stars in two important aspects: (1) helium ignition occurs under degenerate conditions, giving rise to a *helium flash*, and (2) all low-mass stars start helium burning with essentially the same core mass $M_c \ap... | {
"Header 1": "10.3.2 Helium burning in low-mass stars",
"token_count": 1394,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In our $1 M_{\odot}$ example star, the helium flash occurs at point F in Fig. 10.5. Evolution through the helium flash was not calculated for the model shown in this figure. Instead, the evolution of the star is resumed at point G when the helium core has become non-degenerate and has settled into TE with stable He b... | {
"Header 1": "The horizontal branch",
"token_count": 1146,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
During their post-main sequence evolution, stars may undergo one or more episodes during which they are unstable to radial pulsations. The most important manifestation of these pulsations are the *Cepheid* variables, luminous pulsating stars with periods between about 2 and 100 days. It turns out that there is a well-d... | {
"Header 1": "10.4 Pulsational instability during helium burning",
"token_count": 555,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The radial oscillations of a pulsating star result from pressure waves, i.e sound waves that resonate in the stellar interior. These radial oscillation modes are essentially standing waves, with a node at the centre and an open end at the stellar surface – not unlike the sound waves in an organ pipe. Similarly, there a... | {
"Header 1": "10.4.1 Physics of radial stellar pulsations",
"token_count": 917,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In an exactly adiabatic situation the oscillations will maintain the same (small) amplitude. In reality the situation is never exactly adiabatic, which means that the oscillations will generally be damped, unless there is an instability that drives the oscillation, i.e. that makes their amplitude grow.
The requiremen... | {
"Header 1": "10.4.1 Physics of radial stellar pulsations",
"Header 2": "**Driving and damping of pulsations**",
"token_count": 1117,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In stellar envelopes the opacity can be roughly described by a Kramers law, $\kappa \propto \rho T^{-3.5}$ , which when combined with the ideal-gas law implies $\kappa_P \approx 1$ and $\kappa_T \approx -4.5$ . Since for an ionized ideal gas $\nabla_{\rm ad} = 0.4$ , we normally have $\kappa_P + \kappa_T \nabla_{... | {
"Header 1": "The instability strip and the period-luminosity relation",
"token_count": 1174,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The contents of this chapter are also covered by Chapters 25.3.2 and 26.1–26.5 of Maeder, while stellar pulsations and Cepheids are treated in detail in Chapter 15. See also Kippenhahn & Weigert, Chapters 31 and 32.
#### **Exercises**
#### 10.1 Conceptual questions
- (a) Why does the luminosity of a star increase... | {
"Header 1": "Suggestions for further reading",
"token_count": 1070,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
After the central helium burning phase a central core composed of carbon and oxygen is formed. As discussed before, the further evolution of a star differs greatly between massive stars on the one hand, and low- and intermediate-mass stars on the other hand. The evolution of massive stars, in which the core avoids dege... | {
"Header 1": "**Late evolution of low- and intermediate-mass stars**",
"token_count": 219,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The AGB phase starts at the exhaustion of helium in the centre. In the examples of the 5 and 1 *M*<sup>⊙</sup> stars discussed in the previous chapter, this occurs at point H in the evolution tracks (Figs. 10.2 and 10.5). The star resumes its climb along the giant branch, which was interrupted by central helium burning... | {
"Header 1": "**11.1 The asymptotic giant branch**",
"token_count": 232,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
After central He exhaustion the carbon-oxygen core contracts. During a brief transition all layers below the H-burning shell contract (shortly after point H), until He burning shifts to a shell around the CO core. The star now has two active burning shells and a double mirror effect operates: the core contracts, the He... | {
"Header 1": "**The early AGB phase**",
"token_count": 500,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In stars of sufficiently high mass, $M \gtrsim 4\,M_\odot$ (depending somewhat on the initial composition and on whether overshooting is included) a convective dredge-up episode can occur, called the *second dredge-up*. At point K in Fig. 11.1 the convective envelope is seen to penetrate down into the heliumrich laye... | {
"Header 1": "Second dredge-up",
"token_count": 398,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As the He-burning shell approaches the H-He discontinuity, its luminosity decreases as it runs out of fuel. The layers above then contract somewhat in response, thus heating the extinguished H-burning shell until it is re-ignited. Both shells now provide energy and a phase of *double shell burning* begins. However, the... | {
"Header 1": "The thermally pulsing AGB phase",
"token_count": 676,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
After the H-burning shell is reignited, the He-burning shell that lies underneath it becomes geometrically thin. Nuclear burning in such a thin shell is thermally unstable, for the reasons discussed in Sect. 7.5.2. This gives rise to periodic *thermal pulses* of the He-burning shell. What happens during a thermal pulse... | {
"Header 1": "**11.1.1 Thermal pulses and dredge-up**",
"token_count": 1529,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The main effect of thermal pulses and third dredge-up operating in AGB stars is the appearance of helium-burning products at the surface, in particular a large production of carbon. In the 3 *M*<sup>⊙</sup> model shown in Fig. 11.4, the surface <sup>12</sup>C abundance increases after every dredge-up episode and thus g... | {
"Header 1": "**11.1.2 Nucleosynthesis and abundance changes on the AGB**",
"token_count": 494,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Spectroscopic observations show that many AGB stars are enriched in elements heavier than iron, such as Zr, Y, Sr, Tc, Ba, La and Pb. These elements are produced via slow neutron capture reactions on Fe nuclei, the so-called *s-process*. In this context 'slow' means that the time between successive neutron captures is ... | {
"Header 1": "**Production of heavy elements: the s-process**",
"token_count": 807,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Once a star enters the TP-AGB phase it can experience a large number of thermal pulses. The number of thermal pulses and the duration of the TP-AGB phase is limited by (1) the decreasing mass of the H-rich envelope and (2) the growing mass of the degenerate CO core. If the CO core mass is able to grow close to the *Cha... | {
"Header 1": "**Production of heavy elements: the s-process**",
"Header 2": "11.1.3 Mass loss and termination of the AGB phase",
"token_count": 2046,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Nuclear fusion no longer provides energy and white dwarfs shine by radiating the thermal energy stored in their interiors, cooling at almost constant radius and decreasing luminosities. The faintest white dwarfs detected have $L \approx 10^{-4.5} L_{\odot}$ . Observed WD masses are mostly in a narrow range around $0.... | {
"Header 1": "**Production of heavy elements: the s-process**",
"Header 2": "11.1.3 Mass loss and termination of the AGB phase",
"token_count": 1312,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In the interior of a white dwarf, the degenerate electrons provide a high thermal conductivity (Sec. 5.2.4). This leads to a very small temperature gradient, especially because *L* is also very low. The degenerate interior can thus be considered to a have a constant temperature. However, the outermost layers have much ... | {
"Header 1": "**11.2.2 Thermal properties and evolution of white dwarfs**",
"token_count": 2033,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
(11.7), and substituting $c_V = \frac{3}{2} \mathcal{R}/\mu_{\text{ion}}$ , we can write this relation as
$$\tau \approx \frac{1.05 \times 10^8 \,\mathrm{yr}}{\mu_{\mathrm{ion}}} \left(\frac{L/L_{\odot}}{M/M_{\odot}}\right)^{-5/7}.\tag{11.12}$$
This approximate cooling law was derived by Mestel. It shows that more... | {
"Header 1": "**11.2.2 Thermal properties and evolution of white dwarfs**",
"token_count": 562,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The evolution of AGB stars is treated in Chapter 26.6–26.8 of Maeder and Chapter 33.2–33.3 of Kippenhahn & Weigert. White dwarfs are discussed in more detail in Chapter 35 of Kippenhahn & Weigert and Chapter 7.4 of Salaris & Cassisi.
#### **Exercises**
#### 11.1 Core mass luminosity relation for AGB stars
The lum... | {
"Header 1": "Suggestions for further reading",
"token_count": 664,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have seen that low- and intermediate-mass stars (with masses up to $\approx 8\,M_\odot$ ) develop carbon-oxygen cores that become degenerate after central He burning. As a consequence the maximum core temperature reached in these stars is smaller than the temperature required for carbon fusion. During the latest st... | {
"Header 1": "Pre-supernova evolution of massive stars",
"token_count": 541,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Observations in the ultraviolet and infrared part of the spectrum show that luminous stars, with masses above about $15 M_{\odot}$ , undergo rapid mass outflows (stellar winds) that gradually erode their outer layers. An empirical formula that fits the average observed mass-loss rates of stars of roughly solar metalli... | {
"Header 1": "12.1 Stellar wind mass loss",
"token_count": 963,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Cool, luminous stars known as *red supergiants* (RSG) undergo a slow but copious stellar wind that is probably driven by the same mechanism as the 'superwind' of AGB stars: a combination of stellar pulsations and radiation pressure on dust particles that form in the cool outer atmosphere. There are no theoretical predi... | {
"Header 1": "Red supergiant mass loss",
"token_count": 207,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Observations of the most luminous stars in our Galaxy and in the Magellanic Clouds have revealed a clear upper limit to stellar luminosities that depends on the effective temperature (see Fig. 12.2). In particular there are no red supergiants in HR diagram with $\log(L/L_{\odot}) > 5.8$ , which corresponds to the expe... | {
"Header 1": "12.1.1 The Humphreys-Davidson limit and luminous blue variables",
"token_count": 783,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Wolf-Rayet (WR) stars are hot, very luminous stars with bright emission lines in their spectra. The emission indicates very strong, optically thick stellar winds, with mass-loss rates of $\dot{M} \sim 10^{-5} - 10^{-4} M_{\odot}/\text{yr}$ . They are often surrounded by circumstellar nebulae of ejected material. The w... | {
"Header 1": "12.1.1 The Humphreys-Davidson limit and luminous blue variables",
"Header 2": "12.1.2 Wolf-Rayet stars",
"token_count": 359,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Fig. 12.3 shows evolution tracks in the HRD for massive stars calculated with mass loss at metallicity Z = 0.02. As revealed by this figure, the evolutionary journey of a massive star through the HRD can be rather complicated. Evolution proceeds at nearly constant luminosity, because massive stars do not develop degene... | {
"Header 1": "12.2 Evolution of massive stars with mass loss in the HR diagram",
"token_count": 1227,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The evolution of the surface properties described in the previous section corresponds to the hydrogen and helium burning phases of massive stars. Once a carbon-oxygen core has formed after He burning, which is massive enough (> $1.06\,M_\odot$ ) to undergo carbon ignition, the subsequent evolution of the *core* is a s... | {
"Header 1": "12.3 Advanced evolution of massive stars",
"token_count": 577,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In Sect. 6.5 we discussed several weak interaction processes that result in spontaneous neutrino emission at high temperatures and densities, such as photo-neutrinos, plasma-neutrinos and pair annihilation neutrinos. When the central temperature exceeds $\sim 5 \times 10^8$ K, these neutrino losses are the most impor... | {
"Header 1": "12.3.1 Evolution with significant neutrino losses",
"token_count": 888,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
When the temperature in the contracting C-O core reaches $5-8 \times 10^8$ K (depending on the mass of the core), carbon is the first nuclear fuel to be ignited. The reactions involved in carbon burning and further nuclear burning cycles were treated in Sec. 6.4.3. In the following sections we briefly review these an... | {
"Header 1": "12.3.2 Nuclear burning cycles: carbon burning and beyond",
"token_count": 504,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Carbon burning proceeds via the $^{12}\text{C} + ^{12}\text{C}$ reaction, which produces a mixture of products, mainly $^{20}\text{Ne}$ and some $^{24}\text{Mg}$ . Most of the energy produced escapes in the form of neutrinos and only a small fraction is carried away by photons. In stars with masses up to about $2... | {
"Header 1": "**Carbon burning**",
"token_count": 900,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
In stars with masses $\gtrsim 11\,M_{\odot}$ , once the temperature in the contracting O-Ne core reaches $\approx 1.5\times10^9\,\mathrm{K}$ neon is 'burned' into oxygen and magnesium by a combination of photo-disintegration and $\alpha$ -capture reactions (Sec. 6.4.3). Neon burning always occurs in a convective co... | {
"Header 1": "**Carbon burning**",
"Header 2": "Neon and oxygen burning",
"token_count": 654,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
When the central temperature exceeds $3 \times 10^9$ K, a process known as silicon burning starts. Rather than a fusion reaction this is a complex combination of photo-disintegration and $\alpha$ -capture reactions. Most of these reactions are in equilibrium with each other, and their abundances can be described by ... | {
"Header 1": "Silicon burning",
"token_count": 302,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
We have obtained the following general picture of the final stages in the life of a massive star. The C-O core left by helium burning goes through a rapid succession of nuclear burning stages, during which the stellar envelope (and the star's position in the H-R diagram) remains largely unchanged. After the exhaustion ... | {
"Header 1": "12.3.3 Pre-supernova structure",
"token_count": 508,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
#### **12.1 Mass loss of massive stars during the main sequence**
The mass-luminosity relation for massive stars on the main sequence is approximately
$$\log\left(\frac{L}{L_{\odot}}\right) \approx 0.781 + 2.760 \times \log\left(\frac{M_i}{M_{\odot}}\right),$$
where *M<sup>i</sup>* is the initial mass. The mass l... | {
"Header 1": "**Exercises**",
"token_count": 454,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
(a) Assume that all photons transfer their entire *momentum* to the outflowing wind. Show that the maximum mass loss rate that can be driven by radiation is given by
$$\dot{M} < \dot{M}_{\rm max} = \frac{L}{v_{\infty}c},$$
where *v*<sup>∞</sup> is the velocity of the wind at a large distance of the star.
(b) Show... | {
"Header 1": "**Exercises**",
"Header 2": "**12.2 Maximum mass loss rate for a radiation driven wind**",
"token_count": 319,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Supernovae are stellar explosions during which the luminosity of a star reaches $10^9 - 10^{10} L_{\odot}$ at maximum, remaining bright for several months afterward. At least eight supernovae have been observed in our Galaxy over the past 2000 years, by Chinese and in some cases also by Japanese, Korean, Arabian and ... | {
"Header 1": "13.1 Supernovae",
"token_count": 588,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
**Figure 13.1.** Classification of supernovae, based on their spectra and lightcurve shapes. The main supernova types are shown as black squares. Figure from Turatto (2003, LNP 598, 21).
#### **Supernova classification**
On the basis of their spectra, supernovae (SNe) have been histor... | {
"Header 1": "core collapse",
"token_count": 956,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
As indicated in Fig. 13.1, essentially all types of supernova – except Type Ia – appear to be associated with the core collapse of massive stars ( $\gtrsim 8\,M_\odot$ ) at the end of their evolution. The distinction between the different types and subtypes of core-collapse supernovae is related to differences in the s... | {
"Header 1": "13.2 Core collapse and explosion of massive stars",
"token_count": 2007,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The supernova has a luminosity $L\approx 2\times 10^8\,L_\odot$ for up to several months, so that the total energy lost in the form of radiation is $E_{\rm ph}\sim 10^{49}\,{\rm erg}$ . Therefore
$$E_{\rm ph} \sim 0.01 E_{\rm kin} \sim 10^{-4} E_{\rm gr}$$
and $E_{\rm gr} \gg E_{\rm env} + E_{\rm kin} + E_{\rm ph... | {
"Header 1": "13.2 Core collapse and explosion of massive stars",
"token_count": 2048,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
The main physical parameters that determine the appearance of a supernova are:
- the total kinetic energy imparted by the explosion into the envelope,
- the structure (density profile and chemical composition) of the pre-supernova star, as well as the possible presence of circumstellar material lost by the star earli... | {
"Header 1": "13.2.3 Lightcurves of core-collapse supernovae",
"token_count": 1958,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Type Ia supernovae are fundamentally different from other SN types, because they are not associated with the core collapse of massive stars. Instead they are caused by the *thermonuclear explosion* of a CO white dwarf that reaches a critical mass for carbon ignition.
Carbon-burning reactions can occur in a low-temper... | {
"Header 1": "13.3 Type Ia supernovae",
"token_count": 1162,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
#### **13.1 Energy budget of core-collapse supernovae**
- (a) Neutron stars have a radius of about 10 km. Use this to estimate the energy generated during a core collapse supernova (Hint: assume that before the collapse the core is like a white dwarf with mass *M*<sup>c</sup> = *M*Ch, where *M*Ch is the Chandrasekhar... | {
"Header 1": "**Exercises**",
"token_count": 267,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
Silicon burning forms iron out of silicon. Assume that 5 MeV of energy are liberated by creating one <sup>56</sup>Fe nucleus from silicon, and that the final result of this burning is an iron core of about 2 *<sup>M</sup>*⊙. Silicon burning only lasts about one day, as most of the liberated energy is converted into neu... | {
"Header 1": "**13.2 Neutrino luminosity by Si burning**",
"token_count": 379,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/pols11.pdf"
} |
- Astronomy is with mathematics one of the oldest branches of science. It has served as basis for calendars, navigation, has been an important input for religions and was for a long time intertwined with astrology.
- Some of the most important steps in modern astronomy were:
- Galileo performed 1609 the first astronomi... | {
"Header 1": "Astrophysics—some introductory remarks",
"token_count": 1022,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Hipparcos and Ptolemy ( $\approx 150$ B.C.) divided stars into six classes of brightness, called (apparent) magnitudes m: Stars with m=6 were the faintest objects visible by eye, stars with m=1 the brightest stars on the sky. Thus the magnitude scale m increases for fainter objects, a counter-intuitive fact that shoul... | {
"Header 1": "1.1 Brightness of stars",
"token_count": 786,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
A look at Fig. 1.2 shows that the color of stars and galaxies varies considerably. A quantitative way to measure the color of stars is to use filters which sensitivity is centered at different wavelengths λ<sup>i</sup> and compare their relative brightness,
$$m_{\lambda_1} - m_{\lambda_2} = -2.5 \log \frac{\mathcal{F... | {
"Header 1": "1.2 Color of stars",
"token_count": 325,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Planck found empirically 1900 the so-called Kirchhoff-Planck (or simply Planck) distribution
$$B_{\nu} d\nu = \frac{2h\nu^3}{c^2} \frac{1}{\exp(\frac{h\nu}{kT}) - 1} d\nu$$
(1.6)
describing the amount of energy emitted into the frequency interval [ν, ν + dν] and the solid angle dΩ per unit time and area by a body i... | {
"Header 1": "1.3 Black-body radiation",
"Header 3": "1.3.1 Kirchhoff-Planck distribution",
"token_count": 1532,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We can obtain a first estimate of the surface temperature of the Sun from the know sensitivity of the human eye to light in the range 400–700 nm. Assuming that the evolution worked well, i.e. that the human eye uses optimal the light from the Sun, and that the atmosphere is for all frequencies in the visible range simi... | {
"Header 1": "1.3 Black-body radiation",
"Header 3": "Ex.: Surface temperature of the Sun:",
"token_count": 222,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Integrating Eq. (1.6) over all frequencies and possible solid angles gives the energy flux F emitted per surface area A by a black body. The angular integral consists of the solid angle dΩ = dϑ sin ϑdφ and the factor cos ϑ taking into account that only the perpendicular area A<sup>⊥</sup> = A cos ϑ is visible,
$$\int... | {
"Header 1": "1.3 Black-body radiation",
"Header 3": "1.3.3 Stefan-Boltzmann law",
"token_count": 1182,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Relatively nearby stars are seen at slightly different positions on the celestial sphere (i.e. the background of stars that are "infinitely" far away) as the Earth moves around the Sun. Half of this angular difference is called the parallax angle or simply the parallax p. From Fig. 1.6, one can relate the mean distance... | {
"Header 1": "1.3 Black-body radiation",
"Header 3": "1.4 Stellar distances",
"token_count": 523,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The total luminosity L of a star is given by the product of its surface $A=4\pi R^2$ and the radiation emitted per area $\sigma T^4$ ,
$$L = 4\pi R^2 \sigma T^4 \,. \tag{1.22}$$
Since the brightness or energy flux $\mathcal{F}$ was defined as $\mathcal{F} = E/(At) = L/A$ , we recover the inverse-square law fo... | {
"Header 1": "1.3 Black-body radiation",
"Header 3": "1.5 Stellar luminosity and absolute magnitude scale",
"token_count": 811,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
- 1. Show that the mean distance between black-body photons corresponds approximately to the wave-length of photons from the maximum of the Planck distribution, $n_{\gamma} \approx 1/\lambda_{\text{max}}^3$ .
- 2. Derive Wien's law both for frequency and wavelength. Hint: The equations $e^x(5-x) = 5$ and $e^y(3-y) ... | {
"Header 1": "**Exercises**",
"token_count": 267,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Classical astronomy was mainly concerned with the determination of planetary orbits – an understanding of the physical properties of the Sun or stars in general seemed to be impossible. The breakthrough that paved the way to stellar astrophysics was the spectral analysis of stellar light: Already Fraunhofer identified ... | {
"Header 1": "2 Spectral lines and their formation",
"token_count": 210,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Bohr postulated the existence of discrete, stationary orbits for electrons in atoms. More precisely, he required that the angular momentum J = mevr of an electron is a multiple of Planck's constant <sup>~</sup> <sup>≡</sup> h/(2π),
$$J = m_e v r = n\hbar, \qquad n = 0, 1, 2, \dots$$
(2.1)
If the electron is on a bo... | {
"Header 1": "2.1 Bohr-Sommerfeld model for hydrogen-like atoms",
"token_count": 1321,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
A prominent line in the solar spectrum corresponds to the $n=2 \to n=3$ transition and is called Balmer- $\alpha$ or $H\alpha$ . The observation of this line means that a certain fraction of hydrogen in the solar atmosphere must be in the second, excited state. How are the populations of the different excited state... | {
"Header 1": "2.1 Bohr-Sommerfeld model for hydrogen-like atoms",
"Header 3": "2.2 Formation of spectral lines",
"token_count": 1218,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Between 1905 and 1913, Hertzsprung and Russel examined the distribution of stellar luminosities for different spectral classes. The latter quantity characterizes the surface temperature T, and since the luminosity is proportional to T <sup>4</sup> and the stellar surface <sup>∝</sup> <sup>R</sup><sup>2</sup> , one expe... | {
"Header 1": "2.3 Hertzsprung-Russel diagram",
"token_count": 410,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Progress in astronomy has been driven largely by the exploration of new wavelengths ranges of the electromagnetic spectrum. The first discovery of an "extraterrestrial" signal beyond the visible light was the observation of "cosmic rays" by Victor Hess 1912, whose origin is still unresolved. In 1935, Karl Jansky discov... | {
"Header 1": "3 Telescopes and other detectors",
"token_count": 317,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Magnification V : Many objects (binary stars, galaxies) that appear point-like to the naked eye show structure viewed by a telescope. From the left panel of Fig. 3.1, it follows that the magnification V depends on the ratio of the focal lengths f<sup>i</sup> and the apertures D<sup>i</sup> as
$$V = \frac{\phi_2}{\phi... | {
"Header 1": "3.1 Optical telescopes",
"Header 3": "3.1.1 Characteristics of telescopes",
"token_count": 1074,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Chromatic aberration Since the refractive index n of a lens (as any material) is wavelength dependent, a picture of a refracting telescopes cannot be focused on a single common point for different wavelengths. Together with the easier production and support of mirrors, this is the main reason for the dominance of refle... | {
"Header 1": "3.1 Optical telescopes",
"Header 3": "3.1.2 Problems and limitations",
"token_count": 272,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Radio telescopes The simplest realization would be a λ/2 antenna where electromagnetic waves induces resonance currents in it. To increase the sensitivity, one builds large parabolic antenna with the radio receiver in its focal point. Then the signal is electronically amplified.
Since the wavelengths is in the cm to ... | {
"Header 1": "3.2 Other wave-length ranges",
"token_count": 673,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
In principle, all stable particle types may be used to obtain information about the universe: Additionally to photons, these are the charged elementary particles (electrons $e^-$ and protons p) and the stable nuclei. Moreover, the known stable neutral particles are neutrinos $\nu$ and the graviton.
Charged partic... | {
"Header 1": "3.2 Other wave-length ranges",
"Header 3": "3.3 Going beyond electromagnetic radiation",
"token_count": 651,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Consider a light clock, i.e. a light pulse traveling between two mirrors separated by the distance L. The time for a round trip is $t_0 = 2L/c$ . Now consider the clock, i.e. the two mirrors, moving with velocity v, cf. Fig. 4.1. Then the light needs for bouncing once back and forth the time $t > t_0$ ,
$$t^{2} = \... | {
"Header 1": "4 Basic ideas of special relativity",
"Header 3": "4.1 Time dilation – a Gedankenexperiment",
"token_count": 294,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Newton's gravitational law as well as the Coulomb law are examples for an instantaneous action,
$$\mathbf{F}(\mathbf{x}) = K \sum_{i} \frac{\mathbf{x} - \mathbf{x}_{i}}{|\mathbf{x} - \mathbf{x}_{i}|^{3}}.$$
(4.3)
The force $\mathbf{F}(\mathbf{x},t)$ depends on the distance $\mathbf{x}(t) - \mathbf{x}_i(t)$ to a... | {
"Header 1": "4 Basic ideas of special relativity",
"Header 3": "4.2.1 Galilean transformations",
"token_count": 1332,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Another important example for a four-tupel consisting (from a non-relativistic point of view) of a vector and a scalar and transforming like $(ct, \mathbf{x})$ under Lorentz transformations, is $(E, c\mathbf{p})$ . The Lorentz transformation written in this case is
$$E = \gamma (E' + \beta c p_x') \tag{4.18}$$
$... | {
"Header 1": "4 Basic ideas of special relativity",
"Header 3": "4.2.3 Energy and momentum",
"token_count": 352,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We consider as last example how the wave-vector of a photon $(\omega/c, \mathbf{k})$ transforms under a Lorentz transformation,
$$\frac{\omega'}{c} = \frac{\frac{\omega}{c} - \frac{V}{c} k_x}{[1 - (V/c)^2]^{1/2}}.$$
(4.22)
Setting k<sup>x</sup> = k cos α = ω c cos α, where α the angle between the wave vector and ... | {
"Header 1": "4 Basic ideas of special relativity",
"Header 3": "4.2.4 Doppler effect",
"token_count": 831,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
- Apparent or optical binaries are not really binary stars, but random coincidences along the same line-of-sight.
- A visual binary is a bound system that can be resolved into two stars by a telescope. William Herschel discovered 1803 the first example when he noted that the fainter component of Castor (Castor B) had m... | {
"Header 1": "5 Binary stars and stellar parameters",
"Header 3": "Classification of binary stars:",
"token_count": 346,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Kepler developed his three laws of planetary motions empirically from the observations of Tycho Brahe. The laws are
1. Planets orbit the Sun in ellipses, with the Sun in one of the two focuses.
- 2. The line connecting the Sun and a planet sweeps out equal area in equal time.
- 3. The "Harmonic Law" states the squa... | {
"Header 1": "5.1 Kepler's laws",
"token_count": 1049,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Consider the movement of a body under the influence of a central force,
$$\mu \ddot{\mathbf{r}} = f(r) \frac{\mathbf{r}}{r} \,. \tag{5.10}$$
Since $\mathbf{r} \times \mathbf{r} = 0$ , vectorial multiplication with $\mathbf{r}$ should simplify Eq. (5.10). Doing it, we obtain
$$\mu \mathbf{r} \times \ddot{\mathb... | {
"Header 1": "5.1 Kepler's laws",
"Header 3": "5.1.1 Kepler's second law: The area law",
"token_count": 476,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We introduce the unit vector ˆr = r/r and rewrite the definition of the angular momentum L as
$$\mathbf{L} = \mu \mathbf{r} \times \dot{\mathbf{r}} = \mu r \hat{\mathbf{r}} \times \frac{\mathrm{d}}{\mathrm{d}t} (r \hat{\mathbf{r}}) = \mu r \hat{\mathbf{r}} \times \left( \dot{r} \hat{\mathbf{r}} + r \frac{\mathrm{d} \... | {
"Header 1": "5.1 Kepler's laws",
"Header 3": "5.1.2 Kepler's first law",
"token_count": 1138,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Visual binaries The period of a visual binary system can be determined directly, if it is observed long enough. Note that periods are typically long, from years up to 100–1000 yr, because the angular separation of the two stars should be large. The plane containing the orbit may be inclined relative to the plane orthog... | {
"Header 1": "5.2 Determining stellar masses",
"token_count": 1113,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
For any binary system with derived mass also the distance is known. Thus, the luminosity can be derived from the apparent magnitude. When the observed luminosity for stars in binary systems is plotted as function of their mass, the so-called mass-luminosity relationship is obtained. Eddington derived 1924 for main-sequ... | {
"Header 1": "5.2 Determining stellar masses",
"Header 3": "5.2.1 Mass-luminosity relation",
"token_count": 357,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The Sun is a rather typical main-sequence star, distinguished only by its proximity to the observing astronomers on Earth.
**Solar radius** $R_{\odot}$ : From the distance D and the apparent radius of the solar disk, $\delta_{\odot} = 15'59.63''$ , one obtains $R_{\odot} = D\delta_{\odot} = 696000$ km.
**Solar ... | {
"Header 1": "6 Stellar atmospheres and radiation transport",
"Header 3": "6.1 The Sun as typical star",
"token_count": 674,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Let us idealize atoms as black discs of radius R. Then any photon hitting the disc with area $\pi R^2$ is absorbed. In other words, the cross-section $\sigma$ for the absorption of a photon by a single atom is $\sigma = \pi R^2$ .
Consider now a cylinder of length l and area A, filled with N atoms. Their number ... | {
"Header 1": "6.2 Radiation transport",
"token_count": 1387,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
What is the fate of a photon created in the center of the Sun? If we estimate its mean free-path $\ell=1/(n\sigma)$ by assuming that most of hydrogen in the Sun is ionized, and thus the mean density of free electron is $n_e=\rho_\odot/m_H\approx 8.4\times 10^{23}/{\rm cm}^3$ , and use as cross section $\sigma\sim 1... | {
"Header 1": "6.2 Radiation transport",
"Header 3": "6.3 Diffusion and random walks",
"token_count": 742,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Photosphere:** The photosphere gives rise to the continuous radiation and the major part of the Fraunhofer lines. Main process for continuum absorption is the bound-free transition $H^- + \gamma \rightarrow H + e^-$ , for emission the reverse process. Since the electron is free, the energy of the photon is not fixed... | {
"Header 1": "6.4 Photo-, chromosphere and corona",
"token_count": 503,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We denote by M(r) the mass enclosed inside a sphere with radius r and density $\rho(r)$ ,
$$M(r) = 4\pi \int_0^r dr' r'^2 \rho(r')$$
(7.1)
or
$$\boxed{\frac{\mathrm{d}M(r)}{\mathrm{d}r} = 4\pi r^2 \rho(r).}$$
(7.2)
Although trivial, this equation constitutes the first (the "continuity equation") of the five eq... | {
"Header 1": "7 Main sequence stars and their structure",
"Header 3": "7.1.1 Mass continuity and hydrostatic equilibrium",
"token_count": 1990,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Since we use generally the mass density $\rho$ instead of the particle number density n, it is more convenient to introduce the gas constant $R = k/m_H$ and the mean atomic weight $\mu$ defined by $n = \rho/(\mu m_H)$ . Then the ideal gas law becomes
$$P = nkT = R\rho T/\mu.$$
(7.17)
A fully ionized plasma c... | {
"Header 1": "7 Main sequence stars and their structure",
"Header 3": "7.1.1 Mass continuity and hydrostatic equilibrium",
"token_count": 1043,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We want to generalize the virial theorem to a gas with an equation of state such that the pressure P can be expressed as function only of the density $\rho$ . Such an equation of state is called polytropic, and normally written as $P = K \rho^{\gamma}$ with polytropic index $\gamma$ . To do so, we have to express t... | {
"Header 1": "7 Main sequence stars and their structure",
"Header 3": "**7.1.4** \\*\\*\\* **Stability of stars** \\*\\*\\*",
"token_count": 715,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
For the energy flux $\mathcal{F}$ emitted per area and time by an layer of the star at radius r and temperature T, a transport equation similar to Eq. (6.8) for the intensity I holds,
$$d\mathcal{F} = -\sigma n \mathcal{F} dr = -\kappa \rho \mathcal{F} dr. \tag{7.30}$$
Here we introduced also the opacity $\kappa... | {
"Header 1": "7 Main sequence stars and their structure",
"Header 3": "Radiative energy transport",
"token_count": 616,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Thermal equilibrium and energy conservation require that the energy density ǫ produced per time and mass by all possible processes corresponds to an increase of the luminosity L,
$$\frac{\mathrm{d}L}{\mathrm{d}r} = 4\pi r^2 \epsilon \rho \,. \tag{7.35}$$
The energy production rate per time and mass unit, ǫ = dE/(dt... | {
"Header 1": "7 Main sequence stars and their structure",
"Header 3": "7.1.6 Thermal equilibrium and energy conservation",
"token_count": 221,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Our first aim in this section is to derive an upper limit on the luminosity of any stable object. We divide first Eqs. (7.32) and (7.5) and obtain
$$\frac{\mathrm{d}P_{\mathrm{rad}}}{\mathrm{d}P} = \frac{\kappa L(r)}{4\pi c GM(r)} \,. \tag{7.37}$$
Since the total pressure P is the sum of radiation and gas pressure,... | {
"Header 1": "7.2 Eddington luminosity and convective instability",
"token_count": 838,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Converting the hydrostatic equilibrium equation Eq. (7.5) into a difference equation, $dP \rightarrow \Delta P$ and $dr \rightarrow \Delta r$ , gives
$$P \propto M\rho/R$$
. (7.43)
For an ideal gas $P \propto \rho T$ , hence $P \propto M\rho/R \propto \rho T$ or $T \propto M/R$ with $\rho \propto M/R^3$ . ... | {
"Header 1": "7.2 Eddington luminosity and convective instability",
"Header 3": "**7.3.1** Heuristic derivation of $L \\propto M^3$",
"token_count": 241,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Let us define
$$\eta \frac{L}{M} = \frac{L(r)}{M(r)} \tag{7.45}$$
and insert it into Eq. (7.37),
$$\frac{\mathrm{d}P_{\mathrm{rad}}}{\mathrm{d}P} = \frac{\kappa\eta}{4\pi cG} \frac{L}{M} \,. \tag{7.46}$$
At the surface, $\eta=1$ by definition. In general, $\kappa$ increases for small r, while $\eta$ decre... | {
"Header 1": "7.2 Eddington luminosity and convective instability",
"Header 3": "**7.3.2** \\*\\*\\* Analytical derivation of $L \\propto M^3$ \\*\\*\\*",
"token_count": 1713,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Cepheids are named after δ Cephei, a star with a period of 5.4 days and ∆m = 0.7 mag. Generally, the periods of Cepheids vary between P ∼ 1–100 days. H. Leavitt studied 1908–1912 Cepheids in the Small and Large Magellanic Clouds, i.e. at a fixed distance. She discovered a relation between the period and the luminosity ... | {
"Header 1": "7.5 Variable stars – period-luminosity relation of Cepheids",
"token_count": 444,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The mass dM(r) contained in the shell between r and r + dr feels only the gravitational attraction of the enclosed mass M(r). Finding the total gravitational energy of a spherical star amounts therefore to integrate $dU_{pot} = -GM(r)dM(r)/r$ ,
$$U_{\text{pot}}(R) = -G \int_0^R \frac{M(r)dM(r)}{r} \,. \tag{8.1}$$
... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.1.1 Gravitational energy",
"token_count": 805,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The total mass m(Z, N) of a nucleus with Z protons and N neutrons is because of its binding energy $E_b$ somewhat reduced compared to its constituent mass,
$$E_b/c^2 = Zm_H + Nm_n - m(Z, N). (8.3)$$
Measured values of the binding energy per nucleon $E_b/A$ as function of nucleon number A are shown in Fig. 8.1. ... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.1.3 Nuclear fusion",
"token_count": 1006,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Bosons and forces We know four fundamental forces in nature, that are all transmitted by the exchange of bosons, i.e. particles with integer spin. Two of them are the gravitational and electromagnetic force we have already encountered. These two forces fall-off like $1/r^2$ and are important on macroscopic scales. By... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.2 Excursion: Fundamental interactions",
"token_count": 520,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Coulomb barrier-classically** Let us consider the forces between two protons: For $r \gg \lambda_{\pi} = h/(m_{\pi}c)$ , the Coulomb force dominates over the strong force. The size of a nucleon, $r_N \approx 10^{-13}$ cm, is comparable to the Compton wave-length of the pion and we will not distinguish between them... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.3 Thermonuclear reactions and the Gamov peak",
"token_count": 1269,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The pp-chains are shown in detail in in the left panel of Fig. 8.2. Its main chain uses three steps:
Step 1: $p + p \rightarrow d + e^+ + \nu_e$
Step 2: $p + d \rightarrow^3 \text{He} + \gamma$
Step 2: ${}^{3}\text{He} + {}^{3}\text{He} \rightarrow {}^{4}\text{He} + p + p$ .
The CNO-cycle is shown in detail ... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.4.1 Hydrogen burning: pp-chains and CNO-cycle",
"token_count": 1288,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
A "standard solar model" uses the i) conservation laws together with ii) an equation of state, iii) energy production by fusion processes and iv) energy transport equation to evolve the Sun in small time-step from some initial conditions to its present age of $4.5 \times 10^9$ yr. Agreement with present-day luminosit... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.5 Standard solar model and helioseismology",
"token_count": 395,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Solar neutrinos flux** From $L_{\odot} = 4 \times 10^{33} \mathrm{erg/s} = 2 \times 10^{39} \mathrm{MeV/s}$ and the energy release of 26.2 MeV per reaction, the minimal number of neutrinos produced is $\dot{N}_{\nu} = 2 \times 10^{38}/\mathrm{s}$ . As we
Figure 8.4: Schematic pictur... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.6 Solar neutrinos",
"token_count": 1240,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We can consider the time-evolution of a neutrino analogous to the quantum mechanical problem with n states. If we denote the eigenstates of H by $|n\rangle$ , $H|n\rangle = n|n\rangle$ , then their timeevolution is simply
$$|n(t')\rangle = |n(t)\rangle e^{-iE_n(t'-t)}. \tag{8.16}$$
An arbitrary (normalized) state... | {
"Header 1": "8 Nuclear processes in stars",
"Header 3": "8.6.1 \\*\\*\\* Solar neutrino problem and neutrino oscillations \\*\\*\\*",
"token_count": 857,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
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