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In some cases, a star experiences a last thermal pulse while it is moving to the left on the HRD. This is possible because the time between two thermal pulses is on the order of 10<sup>3</sup> to 10<sup>4</sup> yr and the HRD crossing time is of the same order. It is estimated that about one-fourth of all the AGB stars... | {
"Header 1": "19.2 Born-again AGB Stars",
"token_count": 1510,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
When the envelope mass has decreased to $M_{\rm env} < 10^{-4}$ to $10^{-6} M_{\odot}$ , depending on luminosity, shell fusion stops and the luminosity decreases. The star moves down and to the right in the HRD along a *cooling track*. Figure 19.4 shows the cooling tracks from the post-AGB phase via the CSPN phase t... | {
"Header 1": "19.4 Fading to the White Dwarf Phase",
"token_count": 1958,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
All stars with initial masses smaller than about $8M_{\odot}$ end their lives as white dwarfs (WD), degenerate stars with no nuclear energy source. They radiate as they lose thermal energy from their degenerate interior and we can describe their evolution due to this cooling. We will also show that there is a maximum... | {
"Header 1": "White Dwarfs and Neutron Stars",
"token_count": 1702,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The luminosity of WDs comes from cooling. The electrons cannot cool because they are degenerate, so their energy distribution is set by the density, which does not change. Only the ions can cool and they contain almost all of the mass of a WD. Their thermal energy is transported outward by conduction and radiated by th... | {
"Header 1": "20.4 The Cooling of White Dwarfs",
"token_count": 1815,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The equation of state for neutron degeneracy is similar to that of electron degeneracy (Equations (4.20) and (4.22)):
$$P_n = K_{n,1} (\rho/\mu_n)^{5/3}$$
if $\rho \lesssim 6 \times 10^{15} \text{ g cm}^{-3}$ , (20.11)
with $K_{n,1} = 5.5 \times 10^9 \frac{\text{dyne cm}^{-2}}{\left(\text{g cm}^{-3}\right)^{5/3}}... | {
"Header 1": "20.4 The Cooling of White Dwarfs",
"token_count": 1526,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Radially pulsating stars periodically change in size. Their pulsation is due to the existence of a partial ionization zone in the envelope. The location of the classical radial pulsators in the HRD is shown in Figure [21.1.](#page-229-0) Most of these radial pulsators are located in the instability strip, in-between th... | {
"Header 1": "Pulsating Stars",
"Header 3": "21.1 Classical Radial Pulsators",
"token_count": 2016,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
#### 21.3 The $\kappa$ -mechanism of Classical Radial Pulsators
RR Lyrae stars, Cepheids, $\delta$ Scutis, and their Pop II equivalents are variable because the $\kappa$ -mechanism in the H and He ionization zone excites a radial pulsation mode, where $\kappa$ refers to the absorption coefficient. The $\kappa... | {
"Header 1": "Pulsating Stars",
"Header 3": "21.1 Classical Radial Pulsators",
"token_count": 1965,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Let us consider the observed variation of a Cepheid as an example of a radially pulsating star. When a star pulsates radially, it not only changes R but also $T_{\rm eff}$ and L. The bolometric magnitude changes with $R^2T_{\rm eff}^4$ . In general, the star is hotter when it is smaller. The photometric variation at... | {
"Header 1": "21.4 An Example: The Pulsation of δ Cephei",
"token_count": 2036,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Copyright 2002 by the American Physical Society)
From all possible wavelengths, only those that fit with an integer number around the circumference of the star will produce an oscillation. The figure shows the path of a few waves of wavelength *λ* = 2*πR*/ ( ( 1) l l + that produce an oscillation. The pulsations of d... | {
"Header 1": "21.4 An Example: The Pulsation of δ Cephei",
"token_count": 1779,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The distribution of massive stars in the Hertzsprung–Russell diagram is an important source of information about their evolution. The observed luminosity upper limit is not the horizontal line in the HRD, predicted by the Eddington limit, but a line that slopes downward from about 3 × 106 L<sup>ʘ</sup> at ∼40 000 K to ... | {
"Header 1": "Observations of Massive Stars: Evidence for Evolution with Mass Loss",
"token_count": 293,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Figure [22.1](#page-243-1) shows the observed distribution of luminous stars in the LMC with the empirical upper limit. This empirical limit is called the Humphreys–Davidson limit or HD-limit, after the U.S. couple Roberta Humphreys and Kris Davidson, who studied it in 1979 (Humphreys et al. 1979).
The observed distr... | {
"Header 1": "22.1 The Observed Upper Limit in the HRD",
"token_count": 387,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The luminosity of massive stars is so high that they are close to their Eddington limit for radiation pressure (Equation (6.12)). For massive stars, with electron scattering as the dominant opacity in their interior, $L_{\rm E}=4\pi cGM/\sigma_e\approx 3$ to $4\times 10^6L_{\odot}$ and $M_{\rm max}\approx 150$ to... | {
"Header 1": "22.2 The Atmospheric Eddington Limit",
"token_count": 1363,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Luminous Blue Variables (LBVs) are very luminous blue supergiants with L/L<sup>ʘ</sup> > 3 × 10<sup>5</sup> that show large and irregular variations in their V magnitudes (Humphreys & Davidson [1994](#page-255-2)). These variations occur on timescales from weeks to years, with occasional large eruptions.
- − On times... | {
"Header 1": "22.3 Luminous Blue Variables and the Atmospheric Eddington Limit",
"token_count": 1686,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
**Wolf–Rayet stars (WR stars)** are luminous stars, $L \gtrsim 10^5 L_{\odot}$ , with $T_{\rm eff}$ ranging from about ~30,000 to 100,000 K; see Crowther (2007) for a review. They are named after the French astronomers Charles Wolf (1827–1918) and Georges Rayet (1839–1906), who discovered in 1867 a new class of hot ... | {
"Header 1": "22.4 Wolf-Rayet Stars",
"token_count": 1784,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Observations of massive stars in the Galaxy, the LMC, and the SMC show that metallicity plays a crucial role in the evolution of massive stars. This is not surprising considering two effects that we have discussed before.
- 1. The high mass-loss rates of luminous stars are driven by radiation pressure in spectral lin... | {
"Header 1": "22.5 The Dependence of Massive Star Evolution on Metallicity",
"token_count": 888,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. The luminosity upper limit of massive stars, called the Humphreys–Davidson (HD) limit, is a line in the HRD that drops between 50,000 ≳ *T*eff ≳ 9000 K from log L/L<sup>ʘ</sup> ≈ 6.3 to 5.7 and is about constant at *T*eff ≲ 9000 K.
- 2. The atmospheric Eddington limit in the HRD is lower than the classical Eddingt... | {
"Header 1": "22.6 Summary",
"token_count": 1838,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Stars with an initial mass above 8M<sup>ʘ</sup> evolve differently from stars with M < 8M<sup>ʘ</sup> because they are massive enough to go through all nuclear fusion phases (Section [8.11\)](#page-98-2). They do not develop a degenerate He or CO core. By core contraction they can therefore reach the high central tempe... | {
"Header 1": "Evolution of Massive Stars of 8–25M<sup>ʘ</sup>",
"token_count": 303,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Figure [23.1](#page-258-0) shows the predicted evolutionary tracks of stars of Mi = 1 to 120Mʘ, modeled with moderate convective overshooting of ℓos = 0.25 H<sup>p</sup> and mass loss. The left panel is for stars with solar composition (Z = 0.02), and the right panel is for low-metallicity stars (Z = 0.001). The hatche... | {
"Header 1": "23.1 Predicted Evolutionary Tracks",
"token_count": 1204,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Figure [23.2](#page-260-0) shows the Kippenhahn diagram for the evolution of a star of Mi = 15M<sup>ʘ</sup> and metallicity Z = 0.02. The evolutionary track is shown in the left panel of Figure [23.1.](#page-258-0) After the star has arrived on the Hayashi track with an extended convective envelope, it remains an RSG. ... | {
"Header 1": "23.2 The Internal Evolution during the Post-MS Phase of Stars of 8 to 25M<sup>ʘ</sup>",
"token_count": 1764,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. Intermediate-mass stars of 8 ≲ Mi ≲ 25M<sup>ʘ</sup> evolve to the Hayashi line in the post-MS phase. The evolutionary tracks of stars with Mi ≲ 12M<sup>ʘ</sup> describe blue loops in the HRD during core He-fusion. This is due to the contraction of the core and the mirror action of the H-fusion shell.
- 2. The occu... | {
"Header 1": "23.4 Summary",
"token_count": 488,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Mass loss plays a dominant role in the evolution of stars more massive than about $25M_{\odot}$ . During the main-sequence phase, the stars have a convective core that diminishes as the He abundance increases. After the main-sequence phase, stars of $25 \lesssim M_{\rm i} \lesssim 50M_{\odot}$ evolve into red superg... | {
"Header 1": "The Evolution of Massive Stars of $25-120M_{\\odot}$ : Dominated by Mass Loss",
"token_count": 289,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
We have seen in Section 15.2.2 that the mass-loss rate of hot stars increases with luminosity as $\dot{M} \sim L^{1.5}$ . At the same time, the mass-luminosity relation indicates that $L \sim M^y$ with $y \approx 2.9$ so $\dot{M} \sim M^{4.4}$ . This shows that the influence of mass loss on the evolution will inc... | {
"Header 1": "24.1 The Effect of Mass Loss during the Main-sequence Phase",
"token_count": 737,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Figure [24.2](#page-266-0) shows the evolutionary tracks of stars from 0.8 to 120Mʘ, with mass loss and convective overshooting for solar metallicity (Z = 0.014) and SMC metallicity
Figure 24.2. Evolutionary tracks of stars with solar metallicity (Z = 0.014) and SMC metallicity (Z = 0.0... | {
"Header 1": "24.1 The Effect of Mass Loss during the Main-sequence Phase",
"Header 3": "24.2 Predicted Evolution Tracks with Mass Loss",
"token_count": 1546,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Let us consider the evolution of a 60M<sup>ʘ</sup> star as an example of a massive star. Figure [24.4](#page-269-0) shows: the evolutionary track in the HRD (top panel), the changes in internal structure in the Kippenhahn diagram (middle panel), and the changes in the surface composition (lower panel). The letters on t... | {
"Header 1": "24.3 The Evolution of a 60M<sup>ʘ</sup> Star with Mass Loss",
"token_count": 2048,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. The evolution of massive stars is strongly affected by their high mass-loss rates. Mass loss during the MS phase leads to lower luminosities and longer MS lifetimes than those seen in stars with conservative evolution. Mass loss also leads to a widening of the MS in the HRD at high luminosity.
- 2. Mass loss resul... | {
"Header 1": "24.5 Summary",
"token_count": 610,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Massive stars are born rapidly rotating. This is due to the fact that magnetic braking in their early phases is much less important than in lower-mass stars, which have a convective envelope and strong magnetic fields. The distribution of equatorial rotation velocities of massive main-sequence stars shows a peak near a... | {
"Header 1": "Rotation and Stellar Evolution",
"token_count": 296,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Surfaces of constant effective gravity in rapidly rotating stars, where $g_{\rm eff} = g_{\rm grav} - g_{\rm centr}$ , are not spherical but oblate due to the centrifugal acceleration, $g_{\rm centr} = v^2/R$ . The **critical rotation velocity**, $\mathbf{v}_{\rm crit}$ , is defined as the rotation velocity, where ... | {
"Header 1": "25.1 The Critical Velocity of Rotating Stars",
"token_count": 671,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The surface temperature of rapidly rotating stars is not constant but varies with latitude. This can be understood by considering equipotential surfaces. In rapidly rotating stars, surfaces of constant potential are oblate. The potential at a given radius inside a rotating star is
$$\Phi(r,\theta) = -\frac{GM_r}{r} -... | {
"Header 1": "25.2 The Von Zeipel Effect",
"token_count": 785,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The winds from hot stars are driven by radiation pressure. Rapidly rotating stars are more luminous at the pole, where Teff is higher, so they will have a higher mass flux from the pole than from the equator. At the same time, the terminal wind velocity v<sup>∞</sup> will also be higher at the pole than at the equator,... | {
"Header 1": "25.2 The Von Zeipel Effect",
"Header 3": "25.3 Nonspherical Mass Loss of Rapidly Rotating Stars",
"token_count": 1984,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Rapid rotation induces meridional circulation due to shear forces between layers of different rotation speed. Meridional circulation is a flow pattern toward or away from a pole along meridional lines (i.e., lines of constant longitude). This effect is strongest during the main sequence phase when the rotation speed is... | {
"Header 1": "25.4 Mixing by Meridional Circulation",
"token_count": 898,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Rapid rotation affects the evolution of massive stars in two ways.
- Rotation induced mixing will result in a more chemically homogeneous structure than in a nonrotating star. We have seen in Sections [13.2](#page-145-2) that a chemically homogeneous star evolves upward and to the left in the HRD during core H-fusion... | {
"Header 1": "25.4 Mixing by Meridional Circulation",
"Header 3": "25.5 The Effect of Rotation on the Evolution of Massive Stars",
"token_count": 743,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
If the rotation velocity is above a certain limit, mixing can become so strong that the star evolves almost homogeneously, such that at the end of the core H-fusion the star is on the He MS. This limit depends on the metallicity for the following reason. Compared to galactic disk stars, stars of low metallicity have sm... | {
"Header 1": "25.6 Homogeneous Evolution",
"token_count": 239,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. A rapidly rotating star is not spherical but oblate, with a higher temperature at the poles than at the equator: the Von Zeipel effect.
- 2. The winds of rapidly rotating stars are not spherical. In most stars, radiationdriven mass loss is enhanced at the poles due to the higher radiative flux. In some temperature... | {
"Header 1": "25.7 Summary",
"token_count": 1040,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The late evolutionary phases of stars more massive than about Mi ∼ 8M<sup>ʘ</sup> proceed at an increasing speed. This is partly due to the fact that the mass defect ε<sup>n</sup> of the fusion reactions decreases with increasing atomic mass, but at the latest phases of O-fusion and Si-fusion it is also due to the loss... | {
"Header 1": "Late Evolution Stages of Massive Stars",
"Header 3": "26.1 Late Fusion Phases",
"token_count": 2028,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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ª ESO.)
Figure 26.5. Mass of the various regions at the end of the Si-fusion phase as a function of initial stellar mass for stars of solar composition. Left: nonrotating stars. Right: stars with initial rotation velocities of 300 km s<sup>1</sup> . Thick line at top: final mass at the ... | {
"Header 1": "Late Evolution Stages of Massive Stars",
"Header 3": "26.1 Late Fusion Phases",
"token_count": 582,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. Stars with initial masses in excess of ∼12M<sup>ʘ</sup> go through all nuclear fusion and photodisintegration phases. The duration of each next phase is shorter than the previous one, with the phases after C-fusion lasting less than about a year. This is due to strong neutrino losses.
- 2. Because each fusion phas... | {
"Header 1": "Late Evolution Stages of Massive Stars",
"Header 3": "26.4 Summary",
"token_count": 256,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The sudden appearance of bright stars in the sky was attributed in historical times to the visit of a "guest star" or the birth of a new star, hence the name "nova." Later, the intrinsically brightest of these new stars were called "supernovae." Now we know that supernovae are not the beginning but the end of the life ... | {
"Header 1": "Supernovae",
"token_count": 225,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Supernovae are discovered by their sudden brightening. The light curves of two characteristic types of supernovae (SNe), Type Ia and Type II-P, are shown in Figure 27.1. Notice the steep rise in a few tens of days and the slow decline over about a year. The light curve of a Type II-P supernova has a plateau of several ... | {
"Header 1": "27.1 Light Curves of Supernovae",
"token_count": 225,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Stars with an initial mass $M > 12M_{\odot}$ go all the way through Si-fusion and develop an Fe core. Because Fe is the last element that creates energy during its formation by
Figure 27.1. Characteristic light curves of two types of SNe: a thermonuclear SN of Type Ia (red) and core c... | {
"Header 1": "27.2 Core Collapse",
"token_count": 1109,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The **potential energy released during the core-collapse**, when its radius decreases from its initial core radius $R_{ci}$ to its final core radius $R_{cf}$ , is
$$E_{\text{collapse}} \simeq -\frac{GM_c^2}{R_{\text{ci}}} + \frac{GM_c^2}{R_{\text{cf}}} \simeq \frac{GM_c^2}{R_{\text{cf}}} \approx 3 \times 10^{53} \... | {
"Header 1": "27.4 Energetics of Core-collapse Supernovae of Massive Stars",
"token_count": 1929,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
On 1987 February 23, a supernova was detected in the Large Magellanic Cloud (LMC). The blue supergiant Sk $-69^{\circ}202$ of spectral type B3 I that was originally of V = 13.5 magnitude suddenly increased in visual brightness by a factor of $4 \times 10^3$ , reaching V = 4.5 within a day. This was the first time th... | {
"Header 1": "27.4 Energetics of Core-collapse Supernovae of Massive Stars",
"Header 3": "27.6 The Case of Supernova 1987A",
"token_count": 2017,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
An alternative scenario for the evolution of SN 1987A involves the merging of Sk −69°202 during the RSG phase with a lower-mass companion via a common envelope phase. This scenario has the advantage of explaining the unusual abundance of the circumstellar nebula: He/H = 0.15 by number, corresponding to Y = 0.37, and ... | {
"Header 1": "27.4 Energetics of Core-collapse Supernovae of Massive Stars",
"Header 3": "27.6 The Case of Supernova 1987A",
"token_count": 211,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
We have seen above that stars with different initial masses may end their lives differently. Figure [27.6](#page-302-1) shows the final fate of stars with an initial composition of Z = 0.02 as a function ofMi in a schematic way.
- Stars with initial masses less than about 8M<sup>ʘ</sup> have lost a large fraction of ... | {
"Header 1": "27.4 Energetics of Core-collapse Supernovae of Massive Stars",
"Header 3": "27.7 The Remnants of Stellar Evolution",
"token_count": 506,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. Stars with an initial mass ofMi > 8M<sup>ʘ</sup> end their lives with an SN explosion. These SNe are the result of core-collapse when a star has no nuclear fusion source left.
- 2. If the collapsing core has a mass less than about 2Mʘ, the collapse stops when a neutron star is formed. IfM<sup>c</sup> is larger tha... | {
"Header 1": "27.4 Energetics of Core-collapse Supernovae of Massive Stars",
"Header 3": "27.8 Summary",
"token_count": 652,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
A considerable fraction of all stars, possibly as many as half, are born in a binary system. For stars with $M \gtrsim 30~M_{\odot}$ , the fraction might be even higher. The evolution of a star in a binary system will be affected by the presence of the companion if the two stars are within a distance of roughly a few ... | {
"Header 1": "Principles of Close Binary Evolution",
"token_count": 201,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The **period of a binary** in a circular orbit is described by Kepler's third law
$$\left(\frac{2\pi}{P}\right)^2 = \omega^2 = \frac{G(M_1 + M_2)}{a^3},\tag{28.1}$$
where P is the orbital period, $\omega$ is the angular velocity, $M_1$ and $M_2$ are the masses of the two components and a is the separation bet... | {
"Header 1": "28.1 Periods and Angular Momentum",
"token_count": 2043,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The location of the Lagrangian points $L_1$ , $L_2$ , and $L_3$ is indicated. Colors indicate the situation for different diameters of the two components. Green refers to stars that safely fit inside their Roche lobe: matter cannot flow freely from one star to the other. This is called a **detached system**. Blue r... | {
"Header 1": "28.1 Periods and Angular Momentum",
"token_count": 1812,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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Mass transfer occurs when one of the two components fills its Roche lobe. When mass is transferred from a donor to an accretor, three properties must be considered: the change in radius of the donor, the change in radius of the accretor, and the change in separation. They all contribute to a change in the size of the R... | {
"Header 1": "28.5 Stable and Runaway Mass Transfer",
"token_count": 647,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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- 1. The Roche lobe is the smallest equipotential surface that includes both stars. The volume within this 3D surface is the Roche volume.
- 2. When evolution forces a star to increase its size beyond the Roche volume, mass can be transferred to its companion. If the companion can accommodate this mass (i.e., in the ca... | {
"Header 1": "28.6 Summary",
"token_count": 638,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
In this section, we discuss a few examples of close binary evolution with mass transfer. Case A transfer leads the Algol systems and blue stragglers. Case B transfer may lead to a system containing a low-mass Wolf–Rayet star and a more massive O-star. This happens in the case of conservative mass transfer (i.e., all ma... | {
"Header 1": "Close Binaries: Examples of Evolution with Mass Transfer",
"token_count": 271,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
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An Algol system consist of a (sub)giant that fills its Roche lobe and a more massive main-sequence star. The existence of these systems was a puzzle for a long time, since the giant was expected to be more massive because the evolution time is shorter for more massive stars. This was called the Algol paradox. It was so... | {
"Header 1": "29.1 Algol Systems: Conservative Case A Mass Transfer",
"token_count": 1013,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The fraction of main-sequence O stars (luminosity class V) that have one or more bound companions is very high, suggesting that massive stars are often formed in multiple systems (Sana, et al. [2014](#page-326-3)). This implies that binary interactions play a critical role in massive star evolution from the main sequen... | {
"Header 1": "29.2 Massive Interacting Binaries: Conservative Case B Mass Transfer",
"token_count": 1456,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Massive X-ray binaries are the result of merging and spiral-in (Van den Heuvel [1983\)](#page-326-6). As an example of a high-mass X-binary system, we adopt the system LMC X-3. It consists of a B star of 8M<sup>ʘ</sup> and an accreting black hole of 10M<sup>ʘ</sup> in an orbit with a period of 1.7 days. A possible evol... | {
"Header 1": "29.4 The Formation of High-mass X-ray Binaries",
"token_count": 753,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Low-mass X-ray binaries are also the result of binary evolution with spiral-in (Van den Heuvel [1983](#page-326-6)). As an example of a low-mass X-ray binary, we adopt the system Sco X-1, which consists of a neutron star of 1.4M<sup>ʘ</sup> and a low-mass 0.42M<sup>ʘ</sup> companion with an orbital period of 0.88 days.... | {
"Header 1": "29.5 The Formation of Low-mass X-ray Binaries",
"token_count": 765,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Novae are stars that show recurrent outbursts. They are close binary systems with orbital periods of about 1 to 10 hours, consisting of a WD and a star of spectral type G or later with a mass of about 1M<sup>ʘ</sup> or less (Gallagher & Starrfield [1978\)](#page-326-7). There are two types of novae: classical novae and... | {
"Header 1": "29.6 Novae: WDs in Semi-detached Systems",
"token_count": 1868,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The formation of an X-ray binary is not a terminal scenario for a stellar binary system. The subsequent evolution of binaries that include compact objects could in fact lead to more extreme late-time merger scenarios involving compact objects, particularly in the case of massive stars.
In one scenario, it is possible... | {
"Header 1": "29.7 Compact Object Merger Scenarios",
"token_count": 666,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. Algol systems, consisting of a giant and a more massive main-sequence star, are the result of close binary evolution with case A evolution and conservative mass transfer. The primary fills its Roche lobe when it is still in the core H-fusion phase. The receiver is also an MS star and gets more massive than the don... | {
"Header 1": "29.8 Summary",
"token_count": 2052,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
The yields of a star depend on its nuclear evolution and on mass loss by winds or SNe. We therefore briefly summarize the evolution of single stars that were discussed in the previous sections.
0.01 ≲ Mi ≲ 0.8Mʘ: the lifetime of these stars is larger than the age of the Universe, so they are still in their MS phase. ... | {
"Header 1": "30.1 A Summary of the Evolution of Single Stars",
"token_count": 2037,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
This is because O has been converted into C and N in the CNO cycle in low-mass stars.
Figure 30.4 shows the **total yields** for an assumed Salpeter stellar mass function of $N(M_i) = 1000 \times M_i^{-2.35}$ . The large increase in the total yield of He at low mass is due to the rapid increase in the number of thes... | {
"Header 1": "30.1 A Summary of the Evolution of Single Stars",
"token_count": 1654,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. The evolution of the stars described above is far from certain. For instance, meridional circulation, produced by rapid rotation, leads to more and deeper mixing than in slowly rotating stars. Rapid rotation also results in higher mass-loss rates by stellar winds. Moreover, the mass-loss rates in the RSG phase and... | {
"Header 1": "30.1 A Summary of the Evolution of Single Stars",
"Header 3": "Two final comments:",
"token_count": 306,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
- 1. The evolution of stars results in chemical enrichment of the interstellar medium. The amount of enrichment of a given chemical element is called the "chemical yield." It is the mass of an element that is lost by the stellar wind or SN during the lifetime of the star, minus the initial mass of that element in the e... | {
"Header 1": "30.4 Summary",
"token_count": 543,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-0-7503-1278-3.pdf"
} |
Essential Astrophysics is a book to learn or teach from, as well as a fundamental reference for anyone interested in astronomy and astrophysics. This unique volume can be used as a textbook, teaching guide, or reference source for just about anyone interested in astronomy and astrophysics.
It serves as a comprehensiv... | {
"Header 1": "Essential Astrophysics",
"Header 2": "Preface",
"token_count": 2014,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Astronomy is an ongoing, cumulative science in which astronomers either discover previously unseen constituents of the observable universe or determine physical properties of known ones. They measure the mass, luminosity, distance, size, chemical composition, motion, and magnetic fields of planets, stars, galaxies, and... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.1 What Do Astronomers and Astrophysicists Do?",
"token_count": 452,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
In order to observe cosmic objects with any accuracy, we must first establish our bearings here on Earth. In arguments used by Pythagoras (572-479 BC), and subsequently recorded by Aristotle (384-322 BC), it was shown that the Earth is a sphere. During a lunar eclipse, when the Moon's motion carries it through the Eart... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.2 Our Place on Earth",
"token_count": 1443,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
You may have watched the stars as they rise at the horizon on one side of the Earth, slowly move overhead, and eventually set on the other side of the planet, only to reappear the next night. This slow coursing of stars was initially attributed to a revolving celestial sphere, which carried its embedded stars about a s... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.3 Location in the Sky",
"token_count": 1995,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
This sweep of the sky's northern hemisphere was extended to the southern hemisphere by William's son, Sir John Herschel (1792-1871), who published data for 5,079 objects in his General Catalogue in 1864, the combined result of more than half a century of painstaking observations.
Using the Herschel catalogue as a bas... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.3 Location in the Sky",
"token_count": 1153,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Astronomers measure angles in degrees, designated by the superscript , and there are 360 in a circle. They also use the second of arc, or arc second, denoted by the symbol <sup>00</sup>, and the minute or arc, or arc minute, abbreviated by <sup>0</sup> , as a units of angle. The units mimic a clock with 60 s in a minut... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.4 Measuring Angle and Size",
"token_count": 1262,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
While most stars sweep by as the Earth rotates, a star that is aligned with our planet's rotation axis, at the north celestial pole, seems to remain placed in an unchanging location at 90° north declination. The Earth's northern rotation axis, for example, now points close to Polaris, also known as the North Star or th... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.5 The Locations of the Stars are Slowly Changing",
"token_count": 2031,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Because of positional changes caused by precession and nutation, the equinox, or reference date, must be given when specifying the right ascension or declination of any cosmic object. The standard epoch that is now in used for celestial positions is:
$$J2000.0 = 2000 \text{ January } 1.5 = JD2451545.0,$$
where JD... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.5 The Locations of the Stars are Slowly Changing",
"token_count": 2024,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
And in the same way, noon in New York City will occur about 10 min later than in Boston, because New York City is slightly west of Boston.
The world has been divided into standard time zones based on about 1 h, or 15, increments in longitude, so our watches differ from others in hourly increments and are slightly out... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.5 The Locations of the Stars are Slowly Changing",
"token_count": 605,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Astronomers use another sort of time, called sidereal time, to know when and how to point their telescopes to view a particular star or any other cosmic object. The term sidereal is derived from the Latin sidus meaning ''star.'' As with solar time, this star time is based on the Earth's rate of rotation, but measured r... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.7 Telling Time by the Stars",
"token_count": 1101,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Telescopes collect and magnify electromagnetic radiation from a cosmic object, and bigger telescopes provide two advantages. They gather more radiation than a smaller telescope, permitting the detection of fainter objects and providing a brighter image of any cosmic object for analysis. Big telescopes also provide grea... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.8 Optical Telescopes Observe Visible Light",
"token_count": 2020,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Telescopes of different designs are used to detect cosmic radiation outside the optically visible wavelengths. They are used to observe otherwise invisible x-ray, ultraviolet, infrared and radio wavelengths.
Relatively long radio waves are detected by radio telescopes, also known as radio antennas, whose shapes of ar... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.9 Telescopes that Detect Invisible Radiation",
"token_count": 2026,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
It provides information on the formation, composition and evolution of planets, stars and galaxies, and is named after the American astrophysicist Lyman Spitzer, Jr. (1914-1997). Most satellite telescopes, including Hubble, circle our planet outside the Earth's atmosphere while remaining nearby to send observations dow... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.9 Telescopes that Detect Invisible Radiation",
"token_count": 1374,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
By any terrestrial standard, the scale of astronomical objects is enormous in mass, luminosity, distance, size and age. Astronomers and astrophysicists use the Sun's values of these quantities as benchmark units that reflect their large amount. Any solar value is denoted by a subscript symbol $\odot$ , a circle with a... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.10 Units Used by Astronomers and Astrophysicists",
"token_count": 2044,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The nearest large spiral galaxy, Andromeda or M 31, is located at a distance of 0.78 Mpc, while a very remote galaxy might be at a distance of a billion parsec, denoted as a gigaparsec and abbreviated as Gpc, where 1 Gpc = $10^9$ pc.
Astronomers use the second, abbreviated by the lower case letter s, for small time... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.10 Units Used by Astronomers and Astrophysicists",
"token_count": 822,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Since the fundamental laws of physics apply throughout the universe, the physical constants used in the equations that describe these laws are thought to be universal and unvarying in space or time. These constants include the speed of light, c, the Newtonian gravitational constant, designated G, the Boltzmann constant... | {
"Header 1": "Essential Astrophysics",
"Header 2": "1.11 Physical Constants",
"token_count": 1194,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The physical perception of the universe is governed almost solely by the electromagnetic radiation received from cosmic objects. This radiation carries energy and moves through space in periodic waves at the speed of light, designated by the lower case letter c. The speed of light in empty space is a universal constant... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.1 Electromagnetic Waves",
"token_count": 2039,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
\tag{2.9}$$
So, radiation at shorter wavelengths has a higher frequency and a longer wavelength corresponds to a lower frequency. Any electromagnetic wave, regardless of wavelength or frequency, travels though empty space at the speed of light, and it is the maximum speed possible (Focus 2.2).
#### Focus 2.2 Light,... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.1 Electromagnetic Waves",
"token_count": 1051,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Most of us remember the colorful display of a rainbow, which is sunlight bent into separate wavelengths by droplets of water. In the mid-17th century, the English scientist Isaac Newton (1642–1727) showed that sunlight could also be broken into its colors using a prism – a specially cut chunk of glass (Newton 1671, 170... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.2 The Electromagnetic Spectrum",
"token_count": 2030,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Motion changes our perspective, and observations depend on our relative motion with respect to the object being observed. These moving perspectives are described using inertial frames of reference, which move at a constant velocity, never accelerating or decelerating. The Dutch physicist Hendrik A. Lorentz (1853–1928) ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.3 Moving Perspectives",
"token_count": 2022,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
In the *Special Theory of Relativity*, which applies to the non-accelerating and non-gravitational laws of physics, distance is measured by a metric, or line element, *ds*, that combines space, x, y, z, and time, *t*. It was first proposed by Einstein's former teacher, Hermann Minkowski (1864–1909) and is given by Mi... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.3 Moving Perspectives",
"token_count": 391,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
An ideal thermal radiator is known as a blackbody. By definition, a blackbody absorbs all the radiation that falls upon it and reflects none – hence the term black. A black shirt will similarly absorb most of the visible sunlight falling on it and reflects no colors.
Thermal radiation is emitted by a gas in thermal e... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.4 Thermal (Blackbody) Radiation",
"token_count": 2029,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The expression indicates that colder objects radiate most of their energy at longer wavelengths, and that hotter objects are most luminous at shorter wavelengths. In other words, as the temperature of a gas increases, most of its thermal radiation is emitted at shorter and shorter wavelengths.
#### Example: The most ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.4 Thermal (Blackbody) Radiation",
"token_count": 1812,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
When this expression is integrated over all frequencies, we obtain the total energy density, u, of a blackbody:
$$u = \int_{0}^{\infty} u_{\nu}(T)d\nu = aT^{4}, \qquad (2.24)$$
where the radiation constant a is given by
$$a = \frac{8\pi^5 k^4}{15c^3 h^3} \approx 7.57 \times 10^{-16} \,\text{J K}^{-4} \,\text{m}^{... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.4 Thermal (Blackbody) Radiation",
"token_count": 1135,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
How far away is the Sun? The mean distance separating the Earth and the Sun is known as the astronomical unit, abbreviated AU, and it provides the crucial unit of planetary distance. Yet, for a very long time no one knew exactly how big it was. We now know that it is about 149.6 million km.
By the end of the 17th cen... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.5.1 Distance of the Sun",
"token_count": 2020,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
It is approximately 499 s, which corresponds to an AU of about 149.6 million km or $1.496 \times 10^{11}$ m, and approximately 10,000 times the diameter of the Earth. Once scientist's determined the Sun's distance, they could determine the Earth's mean orbital velocity, by assuming – to a first approximation – a circ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.5.1 Distance of the Sun",
"token_count": 1500,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The Sun, like any incandescent body, shines because it is hot. How hot? Once we know the radius and luminosity of the Sun, we can determine the temperature of the Sun's visible disk. Using the Stefan-Boltzmann law, the effective temperature, $T_{\text{eff}(\cdot)}$ , of the visible solar disk is given by:
$$T_{\text... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.5.5 Taking the Sun's Temperature",
"token_count": 1570,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Taking the albedo, A, into account, we have:
$$T_{\rm ep} = 279(1 - A)^{1/4} \left(\frac{\rm AU}{D_P}\right)^{1/2} \rm K.$$
(2.37)
There are two kinds of albedo, the Bond albedo (Bond 1863), which measures the total proportion of electromagnetic energy reflected, and the visual geometric
| Table 2.4 ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.5.5 Taking the Sun's Temperature",
"token_count": 1878,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Keeling's (1928–2005) measurements of the atmospheric carbon dioxide (Arrhenius [1896](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR42); Revelle and Suess [1957;](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR857) Keeling [1960](http://dx.doi.org/10.1007/978-3-642-35963-7_16#CR554), [1978,](http://dx.doi.org/10.... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.5.5 Taking the Sun's Temperature",
"token_count": 441,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
When radiation moves in space from one place to another, it will behave like trains of waves. But when radiation is absorbed or emitted by atoms, it behaves not as a wave but as a package of energy, or like a particle, a photon. A photon is a discrete
60 2 Radiation
quantity of energy associated with electromagneti... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.6 The Energy of Light",
"token_count": 1192,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Our atmosphere is a colorless gas, as you can see in looking at the air in your room, but the sky is usually blue and sunsets are red. The incident sunlight contains all colors, but molecules in our atmosphere scatter blue light from the Sun more than they scatter red sunlight. John Tyndall (1820–1893) discovered the e... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.7.1 Why is the Sky Blue and the Sunsets Red?",
"token_count": 245,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The scattering of radiation by a particle depends on the size, a, of the particle and the wavelength, k, of the radiation. When the particle is much smaller in size than the wavelength, or a k; then the effect is known as Rayleigh scattering, named after Lord Rayleigh (1842–1919). It applies to gas molecules that scatt... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.7.2 Rayleigh Scattering",
"token_count": 2043,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The Compton wavelength for any other subatomic particle is given by the same expression with $m_e$ replaced by the mass of the particle.
In the *inverse Compton effect*, the electrons are not at rest, and may be moving at high speeds. These high-energy electrons scatter low energy photons, and the photons now gain ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.7.2 Rayleigh Scattering",
"token_count": 550,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Once radiation is emitted from an astronomical object, it must pass through intervening space before it reaches the observer. The radiation can be absorbed when passing through a layer or cloud of matter, and the same material can also emit radiation. The material's effect on the radiation is therefore characterized by... | {
"Header 1": "Essential Astrophysics",
"Header 2": "2.7.4 Radiation Transfer",
"token_count": 1435,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Look up at the Sun as it glides across the bright blue sky, or watch the Moon's nightly voyage. On dark, moonless nights you also might notice a bright planet traveling against the stars.
Ancient astronomers thought that the Moon, Sun, and planets all moved in circles, forever wheeling around the central, unmoving Ea... | {
"Header 1": "Essential Astrophysics",
"Header 2": "3.1 Ceaseless, Repetitive Paths Across the Sky",
"token_count": 2049,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
The farther the planet is from the Sun, the weaker the solar force and the slower a planet's motion – as described by Kepler's harmonic relationship.
Roughly half a century later, the great English scientist Isaac Newton (1643–1727) proposed another unseen agent, the invisible gravitational force of the Sun. Newton s... | {
"Header 1": "Essential Astrophysics",
"Header 2": "3.1 Ceaseless, Repetitive Paths Across the Sky",
"token_count": 2041,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
1969; Luther and Towler 1982; Gillies 1997; Fixler et al. 2007). The currently accepted value is:
$$G = 6.67428 \times 10^{-11} \,\mathrm{N} \,\mathrm{m}^2 \,\mathrm{kg}^{-2} = 6.67428 \times 10^{-11} \,\mathrm{m}^3 \,\mathrm{kg}^{-1} \,\mathrm{s}^{-2},$$
(3.3)
with an uncertainty of 1 part in $10^4$ . For computa... | {
"Header 1": "Essential Astrophysics",
"Header 2": "3.1 Ceaseless, Repetitive Paths Across the Sky",
"token_count": 337,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
Assuming a circular orbit at a distance D with period P, the average orbital speed will be $V=2\pi D/P$ , where $\pi=3.14159$ . The mean distance of our Moon from the Earth is $D=384,400~{\rm km}=3.844\times10^8~{\rm m}$ and its orbital period around the Earth is $P=27.3~{\rm Earth}$ days, where 1 day = 86,400 s,... | {
"Header 1": "Essential Astrophysics",
"Header 2": "Example: How fast are the Moon and planets moving, and how do we measure the mass of the planets?",
"token_count": 1999,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
This expression can be used to determine the escape velocity from the gravity of an object of radius R; just equate the kinetic energy $mV^2/2$ to GMm/R to get the velocity V of escape, or $V_{esc}$ , given by:
$$V_{esc} = \left(\frac{2GM}{R}\right)^{1/2},\tag{3.6}$$
which is independent of the small mass m. ... | {
"Header 1": "Essential Astrophysics",
"Header 2": "Example: How fast are the Moon and planets moving, and how do we measure the mass of the planets?",
"token_count": 2012,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/978-3-642-35963-7.pdf"
} |
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