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#### 3.3 Mass of the Sun
The concept of universal gravitation, and Newton's expression for the gravitational force, can be used to derive *Kepler's third law* in the form (see Chap. 4):
$$P_P^2 = \frac{4\pi^2}{G(M_P + M_\odot)} a_P^3, \tag{3.21}$$
where the constant $\pi=3.14159$ , the universal gravitational ... | {
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In this diagram, the height of the tides is greatly exaggerated in comparison to the size of the Earth
Because the Moon's gravitational force decreases with increasing distance, the Moon pulls hardest on the ocean facing it and least on the opposite ocean, whereas the Earth between is pulled with an intermediate forc... | {
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A planet's gravitational force pulls any natural satellite, or moon, into a slightly elongated shape along an axis pointing toward the planet. That is, a planet's gravitation produces two tidal bulges in the solid body of the satellite; one on the closest side to the planet and one on the satellite's farthest side. If ... | {
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As the Earth rotates, the bulge raised on its surface by the Moon's gravity is always a little ahead of the Moon rather than directly under it. The Moon pulls back on the bulge and, in the process, slows down the planet.
When the ocean tides flood and ebb, they create eddies in the water, producing friction on the oc... | {
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According to one of the unbreakable conservation laws, the angular momentum, or the product of mass, M, velocity, V, and radius, R, is unchanged in a closed system, which is not subject to an outside force. Thus:
Conservation of Angular Momentum = $M \times V \times R$ = constant.
This means that the angular mome... | {
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One might expect the numerous particles of a planetary ring to have accumulated long ago into larger satellites. But the interesting feature of these rings – and a clue to their origin – is that they do not coexist with large moons. Planetary rings are also usually closer to the planets than their large satellites.
T... | {
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We cannot see the force of gravity and Newton did not know how it was exerted. Albert Einstein (1879–1955) subsequently explained it by supposing that a massive body like a star bends nearby space. This bending of space is the cause of the star's gravity. However, such effects are noticeable only in extreme conditions ... | {
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[1971\)](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR951), or the time for a radio signal to travel from a spacecraft home (Shapiro et al. [1971;](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR951) Reasenberg et al. [1979](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR842)). When the line of sight passes nea... | {
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[2006;](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR594) Breton et al. [2008\)](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR167). Unlike the binary pulsar PSR B1913 ? 16, this new system contains two pulsars, attributed to two rotating neutron stars that emit radio pulses. They orbit each other at a speed of ... | {
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All that exists, from atoms to planets and stars to galaxies, is always moving. This motion keeps cosmic objects suspended in space.
Galileo Galilei (1564–1642) imagined an ideal world in which there are no external forces acting on an object, and supposed that such an object will keep on moving at constant speed (Ga... | {
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The energy of motion is known as kinetic energy, and for a mass m moving at speed V, the kinetic energy is mV<sup>2</sup> =2, so the faster something moves the more kinetic energy it has. If an object moves fast enough, and its kinetic energy becomes large enough, it can overcome the gravitational forces acting upon it... | {
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But why doesn't the enormous solar
102 4 Cosmic Motion
| Object | Mass (kg) | Radius (m) | Escape speed (km s <sup>-1</sup> ) |
|----------------------------|-------------------------|-----------------------|------------------------------------|
| Ceres, largest asteroid... | {
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When two orbiting objects have comparable mass, as is the case for some binary stars, then the mean orbital velocity, $V_{O1}$ of an object of mass $M_1$ orbiting another mass $M_2$ at a distance a is given by:
$$V_{O1} = \left[ \frac{GM_2^2}{(M_1 + M_2)a} \right]^{1/2}.$$
(4.7)
Here a is the separation of ... | {
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Stars seem to be moving here, there and everywhere, so it is not easy to figure out where they are going. However, a star's motion manifests in two ways, depending on the method used to observe it, and these two components of velocity can be combined to give the direction of motion (Fig. 4.1). The "sideways" velocity c... | {
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"Header 2": "4.3.2 Components of Stellar Velocity",
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The stellar motion that Halley detected is an angular change in a star's position over time, due to its velocity transverse or perpendicular to the line of sight. The angular rate of change is known as proper motion, which is intrinsic to the star and belongs to it, in contrast to any improper motion that might be caus... | {
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"Header 2": "4.3.3 Proper Motion",
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The other component of a star's velocity, the radial velocity directed along the line of sight, can be measured using the Doppler shift of a spectral feature in the star's radiation. Such a feature, called a spectral line, has a definite, well-known wavelength[\(Sect. 6.1\)](http://dx.doi.org/10.1007/978-3-642-35963-7_... | {
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The star with the largest proper motion races across the sky at about 10.4 s of arc, denoted as 10.4", each year. This is *Barnard's star*, named after the American astronomer Edward E. Barnard (1857–1923) who discovered it (Barnard 1916). In our lifetime this star will move by roughly half the angular diameter of the ... | {
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Another type of stellar grouping, known as an open star cluster, includes up to a few thousand stars that were formed at the same time, but are only bound loosely to one another by mutual gravitational attractions (Fig. 4.5). Unlike globular star clusters, which can be held together by its stars' mutual gravitational p... | {
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Some stars race through space with an abnormally high velocity relative to the surrounding interstellar medium. These high-speed stars are known as runaway
Fig. 4.6 Runaway star A high-speed star slams into dense interstellar gas, creating a bow shock wave that may be a million kilometers wide. The star is thought to... | {
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For solid rocky planets, the rotation period is everywhere the same on the planet's surface. The Earth, for example, rotates once very 24 h or 86,400 s at all latitudes, or at every angular distance north or south of the equator. As a result, all points of the globe take the same amount of time to complete one rotation... | {
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It tends to push the equatorial 4.4 Cosmic Rotation 119
| | <u> </u> | | |
|---------|----------------------------|-----------------------------------|-----------------------------------------------... | {
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Observations of sunspots have long indicated that the visible solar disk rotates differently at different latitudes, with a faster rate at the equator than at higher
4.4 Cosmic Rotation 121
| Object | Rotation period | Radiusa (m) | Rotation velocitya (m<br>s-1<br>) |
|-------------------|-----------... | {
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8.5\)](http://dx.doi.org/10.1007/978-3-642-35963-7_8), and the rotation rate inside the Sun has been measured by a change in the periods of the sound waves. Waves propagating in the direction of rotation are carried along by the moving gas, and move faster than they would in a non-rotating Sun. A bird or a jet airplane... | {
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What is matter made of? To find out, we might try breaking any material object into increasingly smaller pieces until we reach a stage when the smallest piece cannot be broken apart. The last step in this imaginary, successive division of matter suggests the existence of unseen atoms, a Greek word meaning ''indivisible... | {
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So electrons and protons are always attracted to each other. Negatively charged electrons surround positively charged protons in an atom, and the total positive charge of the protons is equal to the total negative charge of the electrons. An atom has no net electrical charge and it is electrically isolated from externa... | {
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As in the example of helium, electrically
Table 5.1 Physical properties of electrons, protons, neutrons, and atoms<sup>a</sup>
```
Electron
m_e = \text{mass of electron} = 5.4858 \times 10^{-4} \text{ u} = 9.10938 \times 10^{-31} \text{ kg} = 0.5109989 \text{ MeV}/c^2
e = \text{elementary charge} = 1.6022 \times 10... | {
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The German physician Jules Robert Mayer (1814–1878) reasoned that heat is a form of energy (Mayer [1842](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR689)) – generally called ''force'' in his time and related to the motivating force of fire (Carnot [1824](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR190)). Ma... | {
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At sea level the temperature T is about 288 K, and our air is predominantly composed of nitrogen molecules of mass $m = 2 \times 14 \times u$ , where the atomic mass unit $u = 1.66054 \times 10^{-27}$ kg. Substituting these numbers into our expression for thermal velocity, $V_{thermal} = (3kT/m)^{1/2}$ , where the ... | {
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Also in the mid-nineteenth century, the Scottish physicist James Clerk Maxwell (1831–1879) introduced a statistical approach to the kinetic theory of gases, which recognizes that every gas particle has a different speed and that each collision between particles changes the speeds of those particles. He proposed that th... | {
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The number N(E) of gas particles with a kinetic energy between E and E + dE is given by:
$$N(E)dE = \frac{2N_{tot}}{\pi^{\frac{1}{2}}(kT)^{\frac{3}{2}}} E^{\frac{1}{2}} \exp\left(-\frac{E}{kT}\right) dE = N_{tot}f(E)dE.$$
(5.18)
The total internal energy, U, of a total of $N_{tot}$ particles in thermal equilibriu... | {
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The molecular ingredients of an atmosphere can be determined by observing the unique spectral signatures of different molecules. The atmospheres of the eight major planets in our solar system are mainly composed of molecules of the cosmically abundant atoms – hydrogen, H, carbon, C, oxygen, O, and nitrogen, N, but in v... | {
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Why doesn't the sky fall down, as Chicken Little once said was happening? After all, the Earth's atmosphere is pulled down by the planet's relentless gravity. The answer is that the atmosphere is warmed by the Sun, so its molecules are in continuous motion and collide with one another, producing a gas pressure that pre... | {
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The standard atmospheric pressure on Earth at sea level is equal to $1.01325 \times 10^5$ Pa. The gas temperature is 288 K, so the number density of atmosphere molecules at sea level is $N = N_{tot}/V = P/kT = 2.5 \times 10^{25}$ m<sup>-3</sup>.
The ideal gas law, also known as the perfect gas law, is a simplifie... | {
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Our thin atmosphere is pulled close to the Earth by its gravity and suspended above the ground by molecular motion. And because air molecules are mainly far apart, our atmosphere is mostly empty space, and it always can be squeezed into a smaller volume. The atmosphere near the ground is compacted to its greatest densi... | {
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They showed that there are at least three such reflecting layers, now labeled D, E and F, at respective altitudes of 70, 100 and 200–300 km (Appleton [1932;](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR33) Appleton and Barnett [1925](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR34)).
In 1947, Sir Appleton wa... | {
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5.5 Internal compression of the Sun** The variation of pressure, temperature, and mass density with fractional radial distance from the Sun's center (*left*) to its visible disk (*right*). At the center of the Sun, the temperature is 15.6 million K and the mass density is 151,300 kg m<sup>-3</sup>; the central pressure... | {
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The F layer of the ionosphere, located about 200 km above our heads, contains oxygen atoms that are missing two electrons and a number density of free electrons of $N_e = 10^{12} \text{ m}^{-3}$ . The temperature, T, required to create these ions can be estimated by equating the thermal energy 3kT/2 to the third ioniz... | {
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Whole atoms are only found in the outer visible layers of the Sun, where the temperature is a relatively cool 5,780 K. Raise the temperature by just a factor of three, to about 17,000 K, which happens just beneath the solar disk we see with our eyes, and the Sun's hydrogen atoms are stripped bare, losing their identity... | {
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Sound waves are generated by turbulence (Lighthill 1952, 1954; Proudman 1952), and it is such turbulent motions that give rise to the roar of a jet airplane engine. The convective rise and fall of gas in the outer layers of the Sun also create sound waves (Biermann 1948; Schwarzschild 1948; Schatzman 1949) that are p... | {
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In addition to ponderous material particles, like atoms, molecules, electrons, and ions, there are also magnetic fields that permeate the universe. The trajectories of charged particles, the electrons and the ions, are guided by these magnetic fields,
which act as a wall to them. The charges can spiral around the mag... | {
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When sunlight is spread out into its different colors or wavelengths, it is cut by several dark gaps. They were first noticed by the English astronomer William Hyde Wollaston (1766–1828) in 1802 (Wollaston [1802](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR1136)), and then investigated in greater detail by German ... | {
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Still, there are some very strong absorption lines that are due to the Sun, and they extract large amounts of energy from sunlight. They are produced by hydrogen, sodium, magnesium, calcium, and iron (Table [6.1](#page-182-0)), but iron accounts for more lines than any other element. Because abundant heavy iron accou... | {
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The lines A and B are produced by molecular oxygen in the terrestrial atmosphere
<sup>b</sup> Fraunhofer's D line includes the two sodium lines, designated D1 and D2, and the helium line at 587.6 nm, designated D3
| Element | Symbol | Atomic number, Z | Abundancea (logarithmic) | Discovery on Earth |
|----------... | {
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Most of the mass of an atom is concentrated in its relatively small nucleus, which is surrounded by electrons [\(Sect. 5.1\)](http://dx.doi.org/10.1007/978-3-642-35963-7_5). The nucleus has a positive charge due to the protons in it and is about 100,000 times smaller than the atom. The negatively charged electrons keep... | {
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Because the quantum of an electron's orbital energy increases with the decreasing dimensions of the orbit, this suggested that quantum mechanics would become important only for very small, subatomic physical scales.
In the Bohr atom, the single electron of a hydrogen atom is said to orbit the atom's nuclear proton wi... | {
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The a transition detected at visible wavelengths is called the Balmer a transition, at a red wavelength of 6562.8 Å while the ultraviolet a transition is known as Lyman a at 1215.67 Å, where 1 Å = 10-<sup>10</sup> m.
| The<br>Lyman<br>series, | for<br>n<br>= | 1,<br>includes |
|-------------------------|-----------... | {
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The frequency for the transition from an upper level m to a lower level n is given by
$$v_{mn} = cR_A Z^2 \left(\frac{1}{n^2} - \frac{1}{m^2}\right) \approx 2cR_A Z^2 \frac{(m-n)}{n^3}$$
(6.15)
| Atom | Atomic number, Z | Atomic mass, MA<br>(u) | (107 m-1<br>Rydberg constant, RA<br>) |
|---------------|---... | {
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Because atoms reside together in great numbers, we must use a statistical approach to determine their average properties. Their level of excitation depends on the temperature and the density, which influence how often the particles collide and become excited. The higher the temperature, the faster particles move, and t... | {
"Header 1": "Essential Astrophysics",
"Header 2": "6.3 Some Atoms are Excited Out of Their Lowest-Energy Ground State",
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Because a greater number of atoms will absorb more light, the relative darkness of the absorption lines in the Sun's spectrum should establish the relative abundance of the elements there. That is, darker, stronger absorption lines generally indicate high absorption and therefore larger amounts of the absorbing element... | {
"Header 1": "Essential Astrophysics",
"Header 2": "6.4 Ionization and Element Abundance in the Sun and Other Stars",
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The intensity of a spectral line emitted by atoms or ions in an excited state s will depend on the number density, $N_s$ , of atoms or ions occupying the excited state, the energy $E_s$ of that state, and the temperature T. Under conditions of local thermodynamic equilibrium, the Boltzmann distribution indicates tha... | {
"Header 1": "Essential Astrophysics",
"Header 2": "Example: Intensity of the red hydrogen alpha transition in the solar atmosphere",
"token_count": 1962,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Fowler (1899–1944) and Edward Milne (1896–1950) then showed that the number of atoms or ions responsible for the production of a spectral line can be estimated from the line intensity once the
temperature and pressure of the stellar atmosphere are known (Fowler and Milne [1924\)](http://dx.doi.org/10.1007/978-3-642-3... | {
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For massive, collapsed stars, there also is a detectable gravitational redshift caused by the loss of radiation photon energy in overcoming the immense gravitational pull of a star. This is a small effect for stars like the Sun, whose gravitational redshift is about $2 \times 10^{-6}$ , but it increases for collapsed ... | {
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"Header 2": "6.5.2 Gravitational Redshift",
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Any observed spectral line is the superposition of the lines emitted by many individual atoms in different physical conditions. Rather than appearing at a single wavelength, the observed line therefore is broadened over a range of wavelengths (Fig. 6.8). Van Vleck and Haber (1977) have reviewed absorption and emission ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "6.5.3 Thermal Motion Broadens Spectral Lines",
"token_count": 324,
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The temperature of the visible solar disk, the photosphere, is T=5,780 K, so the thermal velocity, $V_{thermal}$ of a hydrogen atom of mass $m_H=1.007825$ u = $1.6735\times 10^{-27}$ kg, will be $V_{thermal}=(3kT/m_H)^{1/2}=1.20\times 10^4$ m s<sup>-1</sup>, where the Boltzmann constant $k=1.381\times 10^{-23}... | {
"Header 1": "Essential Astrophysics",
"Header 2": "Example: Thermal broadening of atomic hydrogen lines in the photosphere",
"token_count": 233,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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If a source is rotating, the Doppler effect of the object's side rotating toward the observer produces a blueshift to shorter wavelengths; the other side, which is rotating away, shifts a line to longer wavelengths. The combined effect produces a line broadening that increases with the rotation velocity and that depend... | {
"Header 1": "Essential Astrophysics",
"Header 2": "6.5.4 Rotation or Expansion of the Radiating Source can Broaden Spectral Lines",
"token_count": 887,
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When an atom is placed in a magnetic field, it acts like a tiny compass, and it adjusts the energy levels of its electrons. If the atomic compass is aligned in the direction of the magnetic field, the electron's energy increases. If it is aligned in the opposite direction, the energy decreases. Because each energy chan... | {
"Header 1": "Essential Astrophysics",
"Header 2": "6.5.6 Magnetic Fields Split Spectral Lines",
"token_count": 1934,
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The split lines
**Table 6.6** Cosmic magnetic fields
| Object | Magnetic field strength, B (tesla) <sup>a</sup> |
|-----------------------------|-------------------------------------------------|
| Earth (equator to pole) | $3 \times 10^{-5} - 6 \times 10^{-5}$ |
| Solar wind (at ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "6.5.6 Magnetic Fields Split Spectral Lines",
"token_count": 612,
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When voltage is applied to the ends of a glass tube filled with rarefied gas, an arc of light passes from the cathode – negative end – of the gas tube to the other end – the anode, which is positive. If vacuum pumps are used to reduce the gas pressure in the glass tubes, they cease to glow inside, but the glass shines ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.1 The Electron, X-rays and Radium",
"token_count": 1748,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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At the time of their discovery, no one knew exactly what radioactive rays were, where their energy came from, or why the radioactive materials kept pouring out energy. Moreover, the amount of energy being released by radium was difficult to explain, for it far surpassed anything that had been achieved by chemical react... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.2 Radioactivity",
"token_count": 1984,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Why don't the nuclei of the radioactive atoms decay completely all at once? Or to ask a related question, how have the nuclei of so many uranium atoms managed to retain their alpha particles? After all, there is still plenty of uranium around billions of years after the Earth formed, continuing to make the rocks and so... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.3 Tunneling Out of the Atomic Nucleus",
"token_count": 2008,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Radioactive uranium, U238, decays, for example, into lead, Pb206 (which is stable), with a half-life of about 4.47 billion years; so every 4.47 billion years the amount of uranium-238 in a rock will be halved. We can apply the equations to U238, and express the abundance in terms of another kind of lead, $Pb^{204}$ , ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.3 Tunneling Out of the Atomic Nucleus",
"token_count": 2030,
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Although almost all of the neutrinos still would pass through any amount of matter unhindered and undetected, a rare collision with other subatomic particles might leave a trace.
Nuclear reactors produce large numbers of neutrinos, and if a massive detector is placed near a large nuclear reactor, with appropriate shi... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.3 Tunneling Out of the Atomic Nucleus",
"token_count": 674,
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Subatomic particles are entering our atmosphere from all directions in interstellar space and moving at nearly the speed of light. The perpetual high-energy rain was discovered about a century ago, when the Austrian physicist Victor Franz Hess (1883–1964), an ardent amateur balloonist, measured the amount of ionization... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.5 Cosmic Rays",
"token_count": 2024,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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The particle flux is plotted as a function of the particle energy in units of electron volts, abbreviated eV, where $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$ and $1 \text{ GeV} = 10^9 \text{ eV}$ , or 1 billion eV. The most abundant cosmic-ray particles are protons with energies of about $1.5 \times 10^9 \te... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.5 Cosmic Rays",
"token_count": 2008,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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7.4 Cosmic ray **shower** When a primary cosmic ray enters the Earth's atmosphere and collides with the nucleus of an atom in an atmospheric molecule, it can produce a shower of secondary subatomic particles. Here we show the most abundant cosmic ray particle, the proton, producing a neutron, designated n; pions denote... | {
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"Header 2": "7.5 Cosmic Rays",
"token_count": 1891,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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What happens if we turn radioactivity around and instead of watching the nucleus of an unstable heavy atom decay, we bombard a perfectly normal, lighter nucleus with very energetic particles? Perhaps this normally stable element could be transformed artificially on the Earth through such a nuclear bombardment. After al... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.5 Cosmic Rays",
"Header 3": "7.6 Nuclear Transformation by Bombardment",
"token_count": 2023,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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In particle physics, a hadron is a composite particle made of quarks held together by the strong force; the best-known hadrons are the protons and neutrons, which are components of atomic nuclei. CERN is a French acronym for the Conseil Européen pour la Recherche Nucléaire (the European Organization for Nuclear Researc... | {
"Header 1": "Essential Astrophysics",
"Header 2": "7.5 Cosmic Rays",
"Header 3": "7.6 Nuclear Transformation by Bombardment",
"token_count": 1278,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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When we measure the total amount of sunlight that illuminates and warms our globe, and extrapolate back to the Sun, we find that it is emitting an enormous power of 385.4 million, million, million, million, or $3.828 \times 10^{26}$ , watts, where one watt = 1 J s<sup>-1</sup>. This brilliance is far too great to be p... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.1 Can Gravitational Contraction Supply the Sun's Luminosity?",
"token_count": 1980,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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That is we can equate the thermal energy of the proton to the gravitational energy, expressed by the relation:
$$\frac{3}{2}kT_{C\odot} = \frac{Gm_P M_{\odot}}{R_{\odot}},\tag{8.5}$$
where the Boltzmann constant $k=1.38065\times 10^{-23}\,\mathrm{J~K^{-1}}$ , the Newtonian gravitational constant $G=6.674\times 10... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.1 Can Gravitational Contraction Supply the Sun's Luminosity?",
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and radius, $R_S$ , is given by:
#### Table 8.1 Physical properties of the Sun
```
M_{\odot} = \text{mass of Sun} = 1.989 \times 10^{30} \text{ kg}
R_{\odot} = \text{radius of Sun} = 6.955 \times 10^8 \text{ m}
\rho_{\odot} = mean mass density of Sun = 3M_{\odot}/(4\pi R_{\odot}^3) \approx 1{,}409 \text{ kg m}^{-3... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.1 Can Gravitational Contraction Supply the Sun's Luminosity?",
"token_count": 1041,
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The only known method for keeping the Sun shining with its present luminosity for billions of years involves nuclear fusion reactions under the intense pressures and exceptionally high temperatures at great depths within the Sun. They are termed "nuclear" because it is the interaction of atomic nuclei that powers the S... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.1 Mass Lost is Energy Gained",
"token_count": 1931,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Solving for the thermal velocity, $V_{th}$ :
$$V_{th} = \left[\frac{3kT}{m_P}\right]^{1/2} \approx 157 \ T^{1/2} \ \text{m s}^{-1}.$$
(8.16)
For the center of the Sun, where the temperature T=15 million K, the speed is only $V_{th}\approx 6.1\times 10^5$ m s<sup>-1</sup>, more than twenty times lower than that r... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.1 Mass Lost is Energy Gained",
"token_count": 1941,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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The power generated per unit mass, $\varepsilon_{12}$ , by this reaction is given by:
$$\varepsilon_{12} = \frac{r_{12}Q}{\rho} \text{ J s}^{-1} \text{ kg}^{-1}.$$
(8.21)
The total energy released by the proton-proton chain of reactions that make the Sun shine is $Q \approx 26 \text{ MeV} \approx 4 \times 10^{-... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.1 Mass Lost is Energy Gained",
"token_count": 804,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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What is of primary interest in fueling stars is the rate at which the nuclear reactions occur and the power they generate. But before considering these details, it is useful to know the units that nuclear astrophysicists commonly use when considering thermonuclear reactions. The include:
$$\label{eq:Size} \begin{spli... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.2 Understanding Thermonuclear Reactions",
"token_count": 2009,
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The calculation of thermonuclear reaction rates is enormously complex, for it depends upon the specific reaction, whether or not there is a resonance in the reaction cross section, and on accelerator measurements of the reaction, carried out for decades by William A. "Willy" Fowler (1911–1995) and his colleagues; Fow... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.2 Understanding Thermonuclear Reactions",
"token_count": 2033,
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Most of the reactions take place at the effective thermal energy, $E_0$ , or Gamow peak, given by Fowler and Hoyle (1964):
$$E_0 = \left(\frac{kT}{2}\right)^{2/3} E_G^{1/3} = 0.1220 \left(Z_1^2 Z_2^2 A\right)^{1/3} T_9^{2/3} \text{ MeV}$$
(8.33)
where $T_9$ is the temperature in billions of K, or $T_9 = T/10^... | {
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"Header 2": "8.3.2 Understanding Thermonuclear Reactions",
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The detailed nuclear reactions inside stars could not be understood until the 1930s when several subatomic particles were known, including the neutron, discovered in 1932, the positron, detected in cosmic ray showers in 1932, and the neutrino, hypothesized in 1933.
The German physicist Carl Friedrich von Weizsäcker (... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.3 Hydrogen Burning",
"token_count": 1983,
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This energy-producing pair-annihilation reaction can be written symbolically as:
$$e^- + e^+ \to 2\gamma$$
, (8.40)
where each gamma-ray photon has an energy of 0.511 MeV, corresponding to the rest mass energy of an electron, for a total reaction energy release of 1.022 MeV.
The next step follows with little dela... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.3 Hydrogen Burning",
"token_count": 1862,
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That is, the helium nucleus is slightly less massive, by a mere 0.007, or 0.7 %, than the four protons that combine to make it, so there is an energy, $\Delta E$ , released given by:
$$\Delta E = \Delta m c^2 = (4m_p - m_{He})c^2 = 0.007(4m_p)c^2 \approx 4.2 \times 10^{-12} \text{ J}$$
(8.53)
where the mass of the... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.3.3 Hydrogen Burning",
"token_count": 885,
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#### 8.4.1 The Elusive Neutrino
Neutrinos, or "little neutral ones," are tiny, invisible packets of energy with no electrical charge and almost no mass, traveling at nearly the speed of light (see Sect. 7.4). They are produced in great profusion by thermonuclear reactions in the
Sun's core, removing substantial amo... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.4 The Mystery of Solar Neutrinos",
"token_count": 1373,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Unlike a conventional optical telescope, which is placed as high as possible to minimize distortion by the Earth's obscuring atmosphere, a solar neutrino detector is buried beneath a mountain or deep within the Earth's rocks inside mines. This shields the instrument from deceptive signals caused by cosmic rays. There, ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.4.2 Solar Neutrino Detectors Buried Deep Underground",
"token_count": 2003,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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#### Focus 8.5 Leptons
A lepton is an elementary, subatomic particle, whose name comes from the Greek word *lepton* meaning "fine, small, thin, subatomic or slender." Altogether there are six types of leptons, which are divided into two classes: the three charged leptons and the three neutral, or uncharged leptons,... | {
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"Header 2": "8.4.2 Solar Neutrino Detectors Buried Deep Underground",
"token_count": 2032,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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The Sudbury Neutrino Observatory can be operated in two modes: one sensitive only to electron neutrinos and the other equally sensitive to all three types of neutrinos. Observations with both modes have confirmed that the Solar Neutrino Problem is caused by changes in the neutrinos as they travel from the solar core.... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.4.2 Solar Neutrino Detectors Buried Deep Underground",
"token_count": 202,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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All of the Sun's nuclear energy is created deep down inside its high-temperature core, and no energy is created in the cooler regions outside of it. The energygenerating core extends to about one quarter of the distance from the center of the
Fig. 8.5 How Sudbury works Neutrinos from t... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.5 How the Energy Gets Out",
"token_count": 2031,
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When a region of high density is displaced upward into a region of lower density and pressure, convection will take place if the displaced volume expands and becomes less dense than its surroundings. It will then continue to be buoyed up like a balloon or bubbles in a boiling pot of water.
For an adiabatic expansion ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.5 How the Energy Gets Out",
"token_count": 2044,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Roughly 3,000 supergranules are seen on the visible solar disk at any moment. And like the ordinary granulation, the changing pattern of supergranulation is caused by convection. But unlike the granules, whose gases move up and down, the material in each supergranule cell rises in the center, and exhibits a sideways ... | {
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"Header 2": "8.5 How the Energy Gets Out",
"token_count": 1386,
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The Sun has grown slowly in luminous intensity since it formed; with a steady, inexorable brightening that is a consequence of the increasing amount of helium accumulating in the Sun's core. As the hydrogen in the Sun's center slowly depletes, and is steadily replaced by heavier helium, the core must continue producing... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.6 The Faint-Young-Sun Paradox",
"token_count": 798,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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The Sun cannot shine forever, because eventually it will deplete the hydrogen fuel in its core. Although it has converted only a trivial part of its original mass into energy, the Sun has processed a substantial 37 % of its core hydrogen into helium in the past 4.6 billion years. There will be no hydrogen left in the s... | {
"Header 1": "Essential Astrophysics",
"Header 2": "8.7 The Sun's Destiny",
"token_count": 705,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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The apparent edge of the visible solar disk, the photosphere, is illusory, for a hot, transparent atmosphere envelops it, extending all the way to the Earth and beyond. This unseen atmosphere is more rarefied than the best vacuum on Earth, and so tenuous that we see right through it.
The diaphanous outer atmosphere o... | {
"Header 1": "Essential Astrophysics",
"Header 2": "9.1.1 The Million-Degree Solar Corona",
"token_count": 1935,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
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Astronomers have often used the Ångström unit of wavelength, where 1 Ångström = 1 Å = 0.1 nm $\times$ $10^{-10}$ m
| Wavelength (nanometers) | Emitting ion | Formation temperature (kelvin) |
|-------------------------|-----------------|--------------------------------|
| 1.70 | Iron, Fe XVII... | {
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"Header 2": "9.1.1 The Million-Degree Solar Corona",
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The surrounding gas buoys up the concentrated magnetism, and eventually the magnetic fields become strong enough to rise up to the photosphere and break through it in belts of bipolar sunspot pairs (Fig. 9.4).
The initial dipolar magnetic field is twisted into a submerged, ring-shaped field running parallel to the so... | {
"Header 1": "Essential Astrophysics",
"Header 2": "9.1.1 The Million-Degree Solar Corona",
"token_count": 326,
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Magnetic fields are described by lines of force, like those joining the opposite poles of a bar magnet. The direction of the lines of force and the orientation of the magnetic fields can be inferred from the polarization of the spectral lines that have been split by the Zeeman effect. Magnetic-field lines pointing out ... | {
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"Header 2": "9.1.1 The Million-Degree Solar Corona",
"Header 3": "9.1.3 Coronal Loops",
"token_count": 2038,
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The ideal gas law describes it:
$$P_G = N k T, (9.2)$$
where N is the particle number density, k = 1.38065 9 10-<sup>23</sup> J K-<sup>1</sup> is the Boltzmann constant, and T is the temperature. Hotter particles move faster and create greater pressure to oppose the magnetic field, and denser plasma also results in... | {
"Header 1": "Essential Astrophysics",
"Header 2": "9.1.1 The Million-Degree Solar Corona",
"Header 3": "9.1.3 Coronal Loops",
"token_count": 617,
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The visible solar disk, the photosphere, is closer to the Sun's center than the million-degree corona, but the photosphere is several hundred times cooler, with a temperature of 5,780 K. This temperature difference is unexpected because energy should not flow from the cooler photosphere to the hotter corona any more th... | {
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"Header 2": "9.1.4 What Heats the Corona?",
"token_count": 1901,
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In contrast to the dense, bright areas, the corona also contains less dense regions called coronal holes. These so-called holes have so little material in them that they appear as large dark areas on x-ray or extreme-ultraviolet images, seemingly devoid of radiation (also see Fig. [9.8\)](#page-282-0).
The coronal ho... | {
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"Header 2": "9.1.5 Coronal Holes",
"token_count": 263,
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The Sun's radiation is not all that passes through the space between the planets. It is filled with electrons, protons, and magnetic fields emanating from the Sun in a ceaseless flow. These unseen particles and fields form a perpetual solar wind that extends all the way to the Earth and far beyond. It was inferred from... | {
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"Header 2": "9.2.1 The Expanding Sun Envelops the Earth",
"token_count": 2011,
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This radial, supersonic outflow creates a huge bubble of plasma, with the Sun at its center and the planets inside; this is called the *heliosphere*, from *Helios* the "God of the Sun" in Greek mythology.
#### 9.2.2 Properties of the Solar Wind
The million-degree corona is so hot that it cannot stand still. Indeed,... | {
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That seems significant, but in 4.6 billion years the solar wind has only carried away about 0.0001, or one ten thousandth, of the Sun's mass $M_{\odot} = 1.989 \times 10^{30}$ kg. Moreover, that is about three times less than the amount of mass turned into energy during this time by nuclear reactions near the center ... | {
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Instruments aboard spacecraft have detected two solar winds with different physical properties. There is a fast wind that moves at a speed of about 750 km s-<sup>1</sup> and a slow wind that blows at about half that speed. The high-speed wind is steady and uniform, whereas the slow-speed wind is variable and gusty.
T... | {
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"Header 3": "9.2.3 Where Do the Two Solar Winds Come From?",
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How far does the solar wind extend, and where does its influence end? The solar wind carves out a cavity in the interstellar medium known as the heliosphere. Zurbuchen ([2007\)](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR1157) has reviewed the coupling of the Sun and the heliosphere; also see Lang [\(2009](http:/... | {
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