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Sirius is with a distance of 2.6 pc the fifth closest stellar system to the Sun. Analyzing the motions of Sirius 1833–1844, F.W. Bessel concluded that it had an unseen companion, with a orbital period P ∼ 50 yr. In 1862, A. Clark discovered this companion, Sirius B, at apastron or the time of maximal separation of the ... | {
"Header 1": "9.1 Observations of Sirius B",
"token_count": 820,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
For a classical gas, P = nkT, and thus in the limit of zero temperature, also the pressure inside a star goes to zero. How can a star be stabilized after the fusion processes and thus energy production stopped?
The Pauli principle forbids that fermions can occupy the same quantum state. In statistical mechanics, Heis... | {
"Header 1": "9.2 Pressure of a degenerate fermion gas",
"token_count": 974,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Stars like Sirius B that are supported by the pressure of a degenerated electron gas are called white dwarf stars. They have very long cooling times because of their small surface luminosity. This type of stars is rather numerous: The mass density of man-sequence stars in the solar neighborhood is $0.04M_{\odot}/\text... | {
"Header 1": "9.2 Pressure of a degenerate fermion gas",
"Header 3": "9.3 White dwarfs and Chandrasekhar limit",
"token_count": 1450,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Novae and supernovae were characterized empirically according to their luminosity and their spectral lines. Novae show a smaller luminosity increase than supernovae with peak luminosities between 10 and $10^6$ times their average luminosity. They are recursive events with periods in the range $P \sim 1\,\mathrm{h}{-... | {
"Header 1": "9.4 Supernovae",
"token_count": 863,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Generalities of compact stars** White dwarf and neutron stars have in common that their radius is strongly increased with respect to a main sequence star, $R_{\rm WD}/R_{\odot} \sim 10^{-2}$ and $R_{\rm NS}/R_{\odot} \sim 10^{-5}$ . Using two conservation laws involving the stellar radius, we can derive immediatel... | {
"Header 1": "9.4 Supernovae",
"Header 3": "9.5 Pulsars",
"token_count": 1562,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
• Dispersion measure and Galactic electron density: The velocity v of electromagnetic waves in a medium is different from c, $n = v/c \neq 1$ . The refractive index n is a function of the wave-length $\lambda$ and thus there is a dispersion between the arrival times of a pulse at different wave-lengths,
$$\Delta t... | {
"Header 1": "9.4 Supernovae",
"Header 3": "Using pulsars as tool",
"token_count": 284,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
1. In classical mechanics, the equality of gravitating mass m<sup>g</sup> = F/g and inertial mass m<sup>i</sup> = F/a is a puzzle noticed already by Newton. Knowing more forces, this puzzle becomes even stronger. Contrast the acceleration in a gravitational field to the one in a Coulomb field: In the latter, two indepe... | {
"Header 1": "10.1 Basic properties of gravitation",
"token_count": 490,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Consider a freely falling elevator in the gravitational field of a radial-symmetric mass distribution with total mass M. Since the elevator is freely falling, no effects of gravity are felt inside and the space-time coordinates from r = ∞ should be valid inside. Let us call these coordinates K<sup>∞</sup> with x<sup>∞<... | {
"Header 1": "10.2 Schwarzschild metric",
"Header 3": "10.2.1 Heuristic derivation",
"token_count": 975,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Gravitational redshift** As in special relativity, the line-element ds determines the time and spatial distance between two space-time events. The time measured by an observer called the proper-time $d\tau$ is given by $d\tau = cds$ . In particular, the time difference between two events at the same point is given... | {
"Header 1": "10.2 Schwarzschild metric",
"Header 3": "10.2.2 Interpretation and consequences",
"token_count": 1589,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
An accelerated system of electric charges emits dipole radiation with luminosity
$$L_{\rm em} = \frac{2}{3c^3} |\ddot{\mathbf{d}}|^2, \qquad (10.19)$$
where the dipole moment of a system of N charges at position x<sup>i</sup> is d = P<sup>N</sup> <sup>i</sup>=1 qix<sup>i</sup> . One might guess that for the emissio... | {
"Header 1": "10.3 Gravitational radiation from pulsars",
"token_count": 1005,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Black holes thermodynamics** Classically, the mass and therefore the radius of a black hole can only increase with time. The only other quantity in physics with the same property is the entropy, $dS \ge 0$ . This suggests a connection between a quantity characterizing the size of the black hole and its entropy. To d... | {
"Header 1": "10.4 \\*\\*\\* Thermodynamics and evaporation of black holes \\*\\*\\*",
"token_count": 1083,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
- 1. Assume that the surface $A=4\pi R_S^2$ of a Black Hole is emitting black-body radiation with temperature $kT=\hbar c/(4\pi R_S)$ , where $R_S$ is its Scharzschild radius, $R_S=2GM/c^2$ .
- i) Derive the luminosity L of the Black Hole.
- ii) Derive with $E=Mc^2$ and $-\mathrm{d}E/\mathrm{d}t=L$ the lifeti... | {
"Header 1": "10.4 \\*\\*\\* Thermodynamics and evaporation of black holes \\*\\*\\*",
"Header 3": "**Exercises**",
"token_count": 370,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Dust becomes visible by its blocking effect of star light. The combined effect of scattering and absorption of light, mainly by dust grains, is called extinction. Clearly, extinction may affects the properties like the luminosity or the distance of a star or galaxy that we deduce from its observed spectra. Surprisingly... | {
"Header 1": "11.1 Interstellar dust",
"token_count": 985,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Interstellar gas was first detected by its absorption of background light. Since gas clouds are much cooler than the surface of stars, their absorption lines are narrower than those of stars, have smaller excitation energies, and can thus be distinguished from stellar absorption lines.
One expects that gas clouds con... | {
"Header 1": "11.2 Interstellar gas",
"token_count": 435,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
A cloud of gas will collapse, if the gravitational attraction dominates over thermal pressure. A bound system has negative energy $E = E_{pot} + E_{kin} \le 0$ , or
$$\frac{3}{5} \frac{GM^2}{R} \ge \frac{3M}{2m} kT \qquad \text{or} \qquad \frac{M}{R} \ge \frac{5}{2} \frac{kT}{Gm}. \tag{11.7}$$
Here we assumed a ho... | {
"Header 1": "11.2 Interstellar gas",
"Header 3": "11.3.1 Jeans length and mass",
"token_count": 1334,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Luminosity of collapsing clouds** Let us apply once again the virial theorem, $-E_{\rm pot} = 2E_{\rm kin}$ , for the evolution of a collapsing cloud. Its total energy is $E = -E_{\rm kin} = E_{\rm pot}/2$ and as it collapses $|E_{\rm pot}|$ increases, but only half remains as kinetic energy in the cloud. The ot... | {
"Header 1": "11.2 Interstellar gas",
"Header 3": "11.3.2 Protostars",
"token_count": 679,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Types and properties There are two main types of star clusters, galactic or open clusters and globular clusters. Figure 12.1 shows in the left panel the open cluster M45, also called Plejades, and in the right one the globular cluster M80.
- 1. Galactic or open clusters have typically < 10<sup>3</sup> stars and a dia... | {
"Header 1": "12.1 Overview",
"token_count": 1239,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The crucial points in our derivation of the virial theorem $2\langle U_{\rm kin}\rangle = -\langle U_{\rm pot}\rangle$ for a star was the assumption of a gravitationally bound system in equilibrium. Thus it holds also for any other system like a cluster of stars or galaxies, if this system fulfills the two conditions... | {
"Header 1": "12.2 Evolution of a globular cluster",
"token_count": 1971,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
\tag{12.13}$$
An exchange of energy $\delta U_1 = -\delta U_2$ or particles $\delta N_1 = -\delta N_2$ between the two systems leads to a change in the total entropy, $\delta S = \delta S_1 + \delta S_2$ . With U = (3/2)NkT and thus
$$\delta S_i = \frac{\partial S_i}{\partial U_i} \delta U_i + \frac{\partial S... | {
"Header 1": "12.2 Evolution of a globular cluster",
"token_count": 530,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
For a dynamically relaxed cluster, we can use
$$M = \frac{5R\langle v^2 \rangle}{3G} \tag{12.18}$$
to estimate the cluster mass M. Here, $\langle v^2 \rangle$ as before refers to the "thermal" motions of stars. Thus the center-of-mass velocity of the cluster has to be subtracted. Using Doppler shift measurement, ... | {
"Header 1": "12.2 Evolution of a globular cluster",
"Header 3": "12.3 Virial mass",
"token_count": 355,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Stars in a cluster have not only a common distance but were most likely also formed at the same time and from material with the same composition. Studying the Hertzsprung-Russell diagram of stars of the same cluster is therefore possible without knowing the distance to the individual stars. Moreover, a comparison of He... | {
"Header 1": "12.4 Hertzsprung-Russell diagrams for clusters",
"token_count": 326,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
In visible light we see an accumulation of stars as a band across the sky. In the infrared or far-infrared the structure of the Milky Way consisting of a disk and a bulge starts to be revealed. The Milkyway is an example for a spiral galaxy; it can be subdivided into (cf. also Fig. 13.2):
- The galactic bulge with a ... | {
"Header 1": "13 Galaxies",
"Header 3": "13.1 Milky Way",
"token_count": 320,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
In the disk of the Milky Way, stars and other matter is rotating around the center in a regular pattern, as revealed by Doppler effects. In the galactic halo and the galactic bulge, the motion is largely random.
**Determination of the rotation curve** For simplicity, we assume a spherical mass distribution and use ag... | {
"Header 1": "13 Galaxies",
"Header 3": "13.1.1 Rotation curve of the Milkyway",
"token_count": 872,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Supermassive black holes (SMBH) are supposed to be in the center of each galaxy, and that includes our own galaxy, the Milky Way. One way to show the existence of a SMBH is to deduce first the enclosed mass from rotation curves around the supposed BH. Then one has to show that no object with such a mass has a sufficien... | {
"Header 1": "13 Galaxies",
"Header 3": "13.1.2 Black hole at the Galactic center",
"token_count": 381,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Are nebula as e.g. Andromeda galactic objects as globular clusters? Or is our galaxy just one of many in the Universe?
Ernst Julius Öpik estimate 1922 the distance to Andromeda as follows: The rotation velocity at the edge of Andromeda was known from Doppler measurements. Expressing then the radius R of Andromeda as ... | {
"Header 1": "13.2 Normal and active galaxies",
"token_count": 406,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Spiral galaxies are two-dimensional objects, with or without bar. They make up 2/3 of all bright galaxies. They contain gas and dust; young and old stars. The stars move regularly in the disc. The surface luminosity of spiral galaxies decreases exponentially for large radii, L(r) = L<sup>0</sup> exp(−r/r0) with r<sup>0... | {
"Header 1": "13.3 Normal Galaxies",
"Header 3": "13.3.1 Hubble sequence",
"token_count": 277,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Flat rotation curves as far as luminous matter extends are found in practically all galaxies. Hence, as for the Milky way, <sup>v</sup>(r) = const. corresponds to <sup>ρ</sup> <sup>∝</sup> <sup>1</sup>/r<sup>2</sup> , compared to an exponential fall-off of luminous matter.
What are potential explanations for this dis... | {
"Header 1": "13.3 Normal Galaxies",
"Header 3": "13.3.2 Dark matter in galaxies",
"token_count": 1968,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Estimate with <sup>C</sup> = 76, <sup>M</sup> = 5 <sup>×</sup> <sup>10</sup>6M<sup>⊙</sup> and <sup>v</sup> = 220 km/s the maximal distance from which globular clusters could have been spiraled into the center of the galaxy.
ii) The large Magellanic Cloud (LMC), which has <sup>M</sup> = 2×1010M⊙, orbits the Milky Way... | {
"Header 1": "13.3 Normal Galaxies",
"Header 3": "13.3.2 Dark matter in galaxies",
"token_count": 278,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Synchrotron radiation An electron in a homogeneous magnetic field moves on a Larmor orbit with radius r<sup>L</sup> = v⊥mc/(eB). Because of its acceleration, the electron emits electromagnetic radiation. For typical magnetic field strengths found in galaxies, B ∼ µG, and relativistic electrons, the emitted radiation is... | {
"Header 1": "13.4 Active Galaxies and non-thermal radiation",
"Header 3": "13.4.1 Non-thermal radiation",
"token_count": 1030,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Radio galaxies emit typically 10<sup>6</sup> times more energy in the radio range than normal galaxies. The main emission mechanism is synchrotron radiation of electrons. The radio emission comes mainly from two radio lobes separated by up to a distance of order 10 Mpc, cf. Fig. 13.9. Additionally, there is a weaker ra... | {
"Header 1": "13.4 Active Galaxies and non-thermal radiation",
"Header 3": "13.4.2 Radio galaxies",
"token_count": 1014,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Seyfert galaxies** In 1943, Carl Seyfert noticed that certain nearby spiral galaxies have very bright, pinpoint nuclei. The spectra of these galaxies show very strong, often broad, emission lines. The brightness of the cores of Seyfert galaxies fluctuates: The light from the central nucleus varies in less than a year... | {
"Header 1": "13.4 Active Galaxies and non-thermal radiation",
"Header 3": "13.4.3 Other AGN types and unified picture",
"token_count": 896,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Newton's law of universal gravitation changed together with the Copernican principle, i.e. the assumption that all bodies in the Universe obey the same laws as measured on Earth, cosmology from a purely philosophical subject to a sub-discipline of physics. In Newtonian gravity, a static Universe has to be infinite. How... | {
"Header 1": "14 Overview: Universe on large scales",
"Header 3": "14.1 Problems of a static, Newtonian Universe",
"token_count": 265,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Einstein postulated that the Universe is homogeneous and isotropic (at each moment of its evolution). An important question is then at which scale the hierarchical structure of the universe (galaxies, cluster of galaxies, superclusters, ...?) stops.
In the extreme case of a fractal Universe an infinite sequence n of ... | {
"Header 1": "14.2 Einstein's cosmological principle",
"token_count": 557,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**Hubble's law** Hubble found empirically that the spectral lines of "distant" galaxies are redshifted, $z = \Delta \lambda / \lambda_0 > 1$ , with a rate proportional to their distance d,
$$cz = H_0 d. (14.2)$$
If this redshift is interpreted as Doppler effect, $z = \Delta \lambda / \lambda_0 = v_r / c$ , then t... | {
"Header 1": "14.2 Einstein's cosmological principle",
"Header 3": "14.3 Expansion of the Universe: Hubble's law",
"token_count": 1221,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The Table 14.1 summarizes a few methods to measure astronomical distances. The chain of overlapping methods by which astronomers establish distance scales in the universe is called "cosmic distance ladder:" Every extension of the distance ladder inherits all the uncertainties of the previous steps it is based on. It is... | {
"Header 1": "14.2 Einstein's cosmological principle",
"Header 3": "14.3.1 Cosmic distance ladder",
"token_count": 243,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The spatial part dl of the line-element of special relativity,
$$ds^{2} = c^{2}dt^{2} - (dx^{2} + dy^{2} + dz^{2}) = c^{2}dt^{2} - dl^{2},$$
(15.1)
corresponds to the one of an usual euclidean three-dimensional space. Such a space is flat, static, homogeneous and isotropic. An expanding universe means that at diffe... | {
"Header 1": "15.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe",
"token_count": 1173,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Let us consider a sphere of fixed radius at fixed time, dr = dt = 0. The line-element ds simplifies then to $R^2(t)r^2(\sin^2\vartheta d\phi + d\vartheta^2)$ , which is the usual line-element of a sphere $S^2$ with radius rR(t). Thus the area of the sphere is $A = 4\pi (rR(t))^2$ and the circumference of a circle ... | {
"Header 1": "15.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe",
"Header 3": "Geometry of the FRW spaces",
"token_count": 1495,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Let us consider the (Newtonian) energy of a galaxy of mass m at the distance R in a Universe that is very close to homogeneity and isotropy,
$$E = E_{\rm kin} + E_{\rm pot} = \frac{1}{2}m\dot{R}^2 - \frac{GmM(R)}{R}$$
(15.21)
According to Hubble's law we can express the recession velocity $\dot{R}$ of the galaxy ... | {
"Header 1": "15.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe",
"Header 3": "15.2.1 Friedmann equation",
"token_count": 1883,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Additionally to the Friedmann equation we need one equation that describes how the energy content of the Universe is affected by its expansion.
Local energy conservation The first law of thermodynamics becomes with dQ = 0 (no heat exchange to the outside, since no outside exists) simply
$$dU = TdS - PdV = -PdV (15.... | {
"Header 1": "15.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe",
"Header 3": "15.2.2 Local energy conservation and acceleration equation",
"token_count": 1278,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The dependence of different energy forms as function of the scale factor R can derived from energy conservation, dU = -PdV, if an E.o.S. $P = P(\rho) = w\rho$ is specified. For w = const., it follows
$$d(\rho R^3) = -3PR^2 dR \tag{15.41}$$
or eliminating P
$$\frac{\mathrm{d}\rho}{\mathrm{d}R}R^3 + 3\rho R^2 = -... | {
"Header 1": "15.3 Scale-dependence of different energy forms",
"token_count": 604,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We consider a flat universe, k = 0, with one dominating energy component with E.o.S $w = P/\rho = \text{const.}$ . With $\rho = \rho_0 (R/R_0)^{-3(1+w)}$ , the Friedmann equation becomes
$$\dot{R}^2 = \frac{8\pi}{3}G\rho R^2 = H_0^2 R_0^{3+3w} R^{-(1+3w)}, \qquad (15.45)$$
where we inserted the definition of $\r... | {
"Header 1": "15.3 Scale-dependence of different energy forms",
"Header 3": "15.4 Cosmological models with one energy component",
"token_count": 631,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
**General discussion:** We apply now the Friedmann and the acceleration equation to the present time. Thus $\dot{R}_0 = R_0 H_0$ , $\ddot{R} = -q_0 H_0^2 R_0$ and we can neglect the pressure term in Eq. (15.39),
$$\frac{\ddot{R}_0}{R_0} = -q_0 H_0^2 = \frac{\Lambda}{3} - \frac{4\pi G}{3} \rho_{m,0} \,. \tag{15.49}... | {
"Header 1": "15.3 Scale-dependence of different energy forms",
"Header 3": "**15.5** Determining $\\Lambda$ and the curvature $R_0$ from $\\rho_{m,0}$ , $H_0,q_0$",
"token_count": 1773,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
We consider a flat Universe containing as its only two components pressure-less matter and a cosmological constant, Ω<sup>m</sup> + Ω<sup>Λ</sup> = 1. Then the curvature term in the Friedmann equation and the pressure term in the deceleration equation play no role and we can hope to solve these equations for a(t). Mult... | {
"Header 1": "15.6 The ΛCDM model",
"token_count": 1222,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
1. Derive the relation between temperature and time in the early (radiation dominated) universe using ρ = gaT<sup>4</sup> = gπ2T <sup>4</sup>/30 as expression for the energy density of a gas with g relativistic degrees of freedom in the Friedmann equation. What is the temperature at t = 1 s? [Hints: The expression ρ = ... | {
"Header 1": "Exercises",
"token_count": 236,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Different energy form today. Let us summarize the relative importance of the various energy forms today. The critical density $\rho_{\rm cr} = 3H_0^2/(8\pi G)$ has with h=0.7 today the numerical value $\rho_{\rm cr} \approx 7.3 \times 10^{-6} \ {\rm GeV/cm^3}$ . This would corresponds to roughly 8 protons per cubic ... | {
"Header 1": "16.1 Thermal history of the Universe - Time-line of important dates",
"token_count": 1625,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
Nuclear reactions in main sequence stars are supposed to produce all the observed heavier elements up to iron. However, stellar reaction can explain at most a fraction of 5% of $^4$ He, while the production of the weakly bound deuterium and Lithium-7 in stars is impossible. Thus the light elements up to Li-7 are primo... | {
"Header 1": "16.1 Thermal history of the Universe - Time-line of important dates",
"Header 3": "16.2 Big Bang Nucleosynthesis",
"token_count": 2006,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The abundance of $Y(^4\text{He})$ depends mainly on $\exp(-\Delta/T_f)$ : The freeze-out temperature $T_{\rm fr}$ depends in turn on the number of relativistic particles at $t \sim 1$ s and was used as a method to count the number of different light neutrino flavors, cf. right panel of Fig. 16.2. Additionally,... | {
"Header 1": "16.1 Thermal history of the Universe - Time-line of important dates",
"Header 3": "16.2 Big Bang Nucleosynthesis",
"token_count": 367,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
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The left panel of Figure 16.3 shows the distribution of matter obtained in a numerical simulation, while the right panel shows the observed distribution of galaxies from an astronomical survey. On both panels, galaxies are distributed in a honeycomb-like structure: There are voids visible that are essentially free of m... | {
"Header 1": "16.1 Thermal history of the Universe - Time-line of important dates",
"Header 3": "16.3 Structure formation",
"token_count": 538,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
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We discussed earlier that the temperate of photons decreases as $T \sim 1/R$ in an expanding universe. Now we want to justify that the expansion preserves the thermal spectrum in the absence of particle interactions. If reactions like $\bar{f}f \to 2\gamma$ are absent or negligible, the total number N of photons is... | {
"Header 1": "16.1 Thermal history of the Universe - Time-line of important dates",
"Header 3": "Blackbody radiation in an expanding universe",
"token_count": 1378,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
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#### Shortcomings of the standard big-bang model
• Causality or horizon problem: why are even causally disconnected regions of the universe homogeneous, as we discussed for CMB?
The horizon grows like t, but the scale factor in radiation or matter dominated epoch only as $t^{2/3}$ or $t^{1/2}$ , respectively. Th... | {
"Header 1": "16.1 Thermal history of the Universe - Time-line of important dates",
"Header 3": "16.5 Inflation",
"token_count": 1096,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/skript_astro.pdf"
} |
The following integrals frequently appear in the context of calculations involving particle reactions in thermal media, where $\zeta$ refers to the Riemann zeta function.
Table A.1: Thermal integrals.
| | Maxwell-Boltzmann | Fermi-Dirac ... | {
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"Header 3": "A.1 Mathematical formulae",
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Doing only mechanics, the difference between the SI and the cgs system is trivial: The first one uses kg, m and s as basic units, while the latter is based on g, cm and s. Thus derived units like energy, power, etc. differ just by powers of ten. As an example compare the energy units Joule and erg in the two systems: ... | {
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"Header 3": "B.1 SI versus cgs units",
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Gravitational constant $G = 6.674 \times 10^{-11} \text{m}^3 \text{kg}^{-1} \text{s}^{-2} = 6.674 \times 10^{-8} \text{cm}^3 \text{g}^{-1} \text{s}^{-2}$
Planck's constant $\hbar = h/(2\pi) = 1.055 \times 10^{-27} \text{erg s}$
velocity of light $c = 2.998 \times 10^{10} \text{ cm/s}$
Boltzmann constant $k = ... | {
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Astronomical Unit $AU = 1.496 \times 10^{13} \text{ cm}$
Parsec $pc = 3.086 \times 10^{18} \, cm = 3.261 \, ly$ Tropical year $yr = 31556925.2 \, s \approx \pi \times 10^7 \, s$
Solar radius $R_{\odot} = 6.960 \times 10^{10} \text{ cm}$ Solar mass $M_{\odot} = 1.998 \times 10^{33} \text{ g}$ Solar luminosity ... | {
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"Header 3": "**B.4** Astronomical constants and measurements",
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The spectral class, absolute visual magnitude M<sup>V</sup> color index B-V, effective surface temperature radius T, lifetime on the main sequence, and the fraction of the spectral class out of all stars is given in the following table:
| SK | MV | B-V | T/K | lifetime/yr | fraction |
|----|------|-------|-----... | {
"Header 1": "B.7 Properties of main-sequence stars",
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**{**PRIVATE**}**(a). Introduction to Geography
- (b). Elements of Geography
- (c). Scope of Physical Geography
- (d). Geography as an Environmental Science
- (e). History of Physical Geography
- (f). Future of Physical Geography
#### **(a) Introduction to Geography**
**{**PRIVATE**}**The main objective of this o... | {
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In 1847, George Perkins Marsh gave an address to the Agricultural Society of Rutland County, Vermont. The subject of this speech was that human activity was having a destructive impact on land, especially through deforestation and land conversion. This speech also became the foundation for his book **Man and Nature** o... | {
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It suggested instead that continuing uniformity of existing processes were responsible for the present and past conditions of this planet.
- **(2).** Evolution Charles Darwin's Origin of Species (1859) suggested that natural selection determined which individuals would pass on their genetic traits to future generations... | {
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#### **(a) Introduction to Maps**
#### **{**PRIVATE**}Introduction**
A map can be simply defined as a graphic representation of the real world. This representation is always an abstraction of reality. Because of the infinite nature of our Universe it is impossible to capture all of the complexity found in the real ... | {
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Graphic scales are quite useful because they can be used to measure distances on a map quickly.
**{**PRIVATE**}Figure 2a-9:** The following graphic scale was drawn for map with a scale of 1:250,000. In the illustration distances in miles and kilometers are graphically shown.
Maps are o... | {
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In this modification, the six degree wide zones are divided into smaller pieces or quadrilaterals that are eight degrees of latitude tall. Each of these rows is labeled, starting at 80 degrees South, with the letters C to X consecutively with I and O being omitted (**Figure 2b-5**). The last row X differs from the othe... | {
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Due south has an azimuth of 180 degrees.
**{**PRIVATE**}Figure 2b-9:** *Azimuth* system for measuring direction is based on the 360 degrees found in a full circle. The illustration shows the angles associated with the major cardinal points of the compass. Note that angles are determined ... | {
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Pipeline, multiple pipelines, control valve
| Pipeline, underground multiple pipelines, underground | |
|-------------------------------------------------------|--------|
| Electric facility | |
| Power transmission line multiple lines | |
| Tele... | {
"Header 1": "**Human Activity Symbols - Recreation {**PRIVATE**}Feature Name Symbol** Sports track Swimming pool Stadium Golf course Golf driving range Campground; Picnic site Ski area, ski jump Rifle range with butts Historic site or point of interest; Navigation light Aerial cableway, ski lift **Human Activity Sy... |
The simplest form of remote sensing uses photographic cameras to record information from **visible** or **near** *infrared* wavelengths (**Table 2e-1**). In the late 1800s, cameras were positioned above the Earth's surface in balloons or kites to take *oblique aerial photographs* of the landscape. During World War I... | {
"Header 1": "**Human Activity Symbols - Recreation {**PRIVATE**}Feature Name Symbol** Sports track Swimming pool Stadium Golf course Golf driving range Campground; Picnic site Ski area, ski jump Rifle range with butts Historic site or point of interest; Navigation light Aerial cableway, ski lift **Human Activity Sy... |
In France, the *SPOT* (**Centre National d'Etudes Spatiales**) satellite program has launched four satellites since 1986. Since 1986, SPOT satellites have produced more than 5.5 million images. SPOT satellites use two different sensing systems to image the planet. One sensing system produces black and white panchromati... | {
"Header 1": "**Human Activity Symbols - Recreation {**PRIVATE**}Feature Name Symbol** Sports track Swimming pool Stadium Golf course Golf driving range Campground; Picnic site Ski area, ski jump Rifle range with butts Historic site or point of interest; Navigation light Aerial cableway, ski lift **Human Activity Sy... |
In the 1980s and 1990s, many GIS applications underwent substantial evolution in terms of features and analysis power. Many of these packages were being refined by private companies who could see the future commercial potential of this software. Some of the popular commercial applications launched during this period ... | {
"Header 1": "**Human Activity Symbols - Recreation {**PRIVATE**}Feature Name Symbol** Sports track Swimming pool Stadium Golf course Golf driving range Campground; Picnic site Ski area, ski jump Rifle range with butts Historic site or point of interest; Navigation light Aerial cableway, ski lift **Human Activity Sy... |
#### **(a) Scientific Method**
**{**PRIVATE**}***Francis Bacon* (1561-1626), a 17th century English philosopher, was the first individual to suggest a universal methodology for *science*. Bacon believed that scientific method required an *inductive* process of inquiry. *Karl Popper* later refuted this idea in the 20t... | {
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Out of this research Physical Geographers determined such things as: the climatic characteristics for specific locations and regions of the planet; flow rates of rivers; soil characteristics for various locations on the Earth's surface; distribution ranges of plant and animal species; and calculations of the amount of ... | {
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The first step in the calculation of *standard deviation* is to determine the *variance* by obtaining the deviations of the individual values (Xi) from the mean $(\overline{X})$ . The formula for *variance* ( $S^2$ ) is:
$$S^2 = [\Sigma(Xi - \overline{X})^2]/(N-1)$$
where $\Sigma$ is the summation sign, $(Xi ... | {
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These values are then entered in the formulae shown under **Table 3g-1** for the calculation of **U** and **U1**.
**{**PRIVATE**}Table 3g-1:** Analysis of convective precipitation levels per storm event (mm of rain) between urban and rural areas using the Mann-Whitney U test.
| {PRIV<br>ATE}U | Rural (n2) | Ra... | {
"Header 1": "**3) The Science of Physical Geography**",
"Header 2": "**S r1** and **S r2**",
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**{**PR IVA TE**}** n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 - - - - - - - - - - - - - - - - - - - - 2 - - - - - - - 0 0 0 0 1 1 1 1 1 2 2 2 2 3 - - - - 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 4 - - - - - - 3 4 4 5 6 7 8 9 10 11 11 12 13 13 5 - 0 1 2 2 3 5 6 7 8 9 10 12 13 14 15 17 18 19 20 6 - - - - - 5 6 8 10... | {
"Header 1": "**3) The Science of Physical Geography**",
"Header 2": "**S r1** and **S r2**",
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| {PRIVATE}Cont<br>rol (Xa) | (Xa)2 | Engineered (Xb) | (Xb)2 |
|---------------------------|---------|------------------|---------|
| 10.7 | 114.49 | 10.0 | 100 |
| 6.7 | 44.89 | 10.2 | 104.04 |
| 8.7 | 75.69 | 12... | {
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"Header 2": "**S r1** and **S r2**",
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| {PRIVATE}Degrees<br>of Freedom | P=0.10 | P=0.05 | P=0.02 | P=0.01 | P=0.001 | Degrees of Freedom |
|--------------------------------|--------|--------|--------|--------|---------|--------------------|
| 1 | 6.314 | 12.706 | 31.821 | 63.657 | 636.619 | 1 |
| 2 ... | {
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"Header 2": "**S r1** and **S r2**",
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Obtaining a significant calculated F-value indicates that the results of regression and correlation are indeed true and not the consequence of chance.
#### **Simple Linear Regression**
In a simple regression analysis, one *dependent variable* is examined in relation to only one *independent variable*. The analysis ... | {
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"Header 2": "**S r1** and **S r2**",
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{PRIVATE}**Figure 3h-2**: Scattergram plot of the precipitation and cucumber yield data and the regression model best fit straight-line describing the linear relationship between the two variables.
#### Regression Analysis and ANOVA
A regression model can be viewed of as a type of m... | {
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"Header 2": "**S r1** and **S r2**",
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#### **(a) Humans and Their Models**
**{**PRIVATE**}**The world of nature is very complex. In order to understand this complexity humans usually try to visualize the phenomena of nature as a *system*. A system is a set of interrelated components working together towards some kind of process. One of the simplest forms... | {
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"Header 2": "**4) Introduction to Systems Theory**",
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We will learn more about these laws later in this textbook.
#### **(e) Food Chain as an Example of a System**
**{**PRIVATE**}**A *food chain* models the movement of *energy* in an *ecosystem* (a form of environmental system). **Figure 4e-1** below illustrates the movement of energy in a typical food chain. In this ... | {
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#### **(a) Evolution of the Universe**
**{**PRIVATE**}**About 10 to 20 billion years ago all of the matter and energy in the *Universe* was concentrated into an area the size of an atom. At this instant, matter, energy, space and time did not exist. Then suddenly, the Universe began to expand at an incredible rate an... | {
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"Header 2": "**5) The Universe, Earth, Natural Spheres, and Gaia**",
"token_count": 2041,
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*Hydrosphere* - describes the waters of the Earth (see the *hydrologic cycle*). Water exists on the Earth in various stores, including the *atmosphere*, *oceans*, *lakes*, *rivers*, *soils*, *glaciers*, and *groundwater*. Water moves from one store to another by way of: *evaporation*, *condensation*, *runoff*, *preci... | {
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"Header 2": "**5) The Universe, Earth, Natural Spheres, and Gaia**",
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#### **(a) Characteristics of Energy and Matter**
#### **{**PRIVATE**}Introduction**
*Energy* is defined simply by scientists as the capacity for doing work. *Matter* is the material (*atoms* and *molecules*) that constructs things on the Earth and in the Universe. Albert Einstein suggested early in this century th... | {
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**Metals** are elements that usually conduct heat and electricity and are shiny. **Nonmetals** do not conduct electricity that well and are normally not shiny. **Metalloids** have characteristics that are in between metals and nonmetals.
Elements with a net positive or negative charge are called *ions*. Chemists indi... | {
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The secretory vesicles are then<br>transported to the cell surface where they are release to<br>the environment outside the cell. |
| Vacuole | Voids within the cytoplasm. Quite large in plant cells. | Used to stor... | {
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"token_count": 1987,
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Sugars created in photosynthesis can be later converted by the plant to starch for storage, or it can be combined with other sugar molecules to form specialized carbohydrates such as *cellulose*, or it can be combined with other nutrients such as nitrogen, phosphorus, and sulfur, to build complex molecules such as *pro... | {
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Using this law we can model the effect that distance traveled has on the intensity of emitted radiation from a body like the sun. **Figure 6f-4** suggests that the intensity of radiation emitted by a body quickly diminishes with distance in a nonlinear fashion.
**{**PRIVATE**}Figure 6f-... | {
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"token_count": 2019,
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**{**PRIVATE**}Figure 6h-3:** During the summer solstice the Earth's North Pole is tilted 23.5 degrees towards the sun relative to the circle of illumination. This phenomenon keeps all places above a latitude of 66.5 degrees N in 24 hours of sunlight, while locations below a latitude of... | {
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**Figure 6h-10:** Solar noon sun angles for 60 degrees N.
**Figure 6h-11:** Solar noon sun angles for 23.5 degrees N.
**Figure 6h-12:** Solar noon sun angles for the equator
**Figure 6h-13:** ... | {
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#### **(a) Atmospheric Composition**
**{**PRIVATE**}Table 7a-1** lists the eleven most abundant gases found in the Earth's lower atmosphere by volume. Of the gases listed, nitrogen, oxygen, water vapor, carbon dioxide, methane, nitrous oxide, and ozone are extremely important to the health of the Earth's biosphere. ... | {
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It is also the layer in the atmosphere where the *jet streams* occur.
Above the tropopause, is the *stratosphere*. This layer extends from an average altitude of 20 to 48 kilometers above the Earth's surface. In the stratosphere, temperature increases with altitude because a localized
concentration of *ozone* gas m... | {
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"token_count": 2007,
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The layer sits at an altitude of about 10 to 50 kilometers, with a maximum concentration in the *stratosphere* at an altitude of approximately 25 kilometers. In recent years, scientists have measured a seasonal thinning of the ozone layer primarily at the South Pole. This phenomenon is being called the *ozone hole*.
... | {
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(Image produced by the *CoVis Greenhouse Effect Visualizer*).
The *reflectivity* or *albedo* of the Earth's surface varies with the type of material that covers it. For example, fresh snow can reflect up to 95 % of the insolation that reaches it surface. Some other surface type reflectivities are:
Dry sand 35 to 45... | {
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Over 90 % of this emission of longwave energy is directed back to the Earth's surface where it once again is absorbed by the surface. The heating of the ground by the longwave radiation causes the ground surface to once again radiate, repeating the cycle described above, again and again, until no more longwave is avail... | {
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"token_count": 1898,
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{PRIVATE}Shortwave radiation from the sun enters the surface-atmosphere system of the Earth and is ultimately returned to space as longwave radiation (because the Earth is cooler than the sun). A basic necessity of this energy interchange is that incoming solar insolation and outgoing radiation be equal in quantity. On... | {
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"Header 2": "(i) Net Radiation and the Planetary Energy Balance",
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Cold surfaces have low values of loss. Color range: white - red - blue, Values: -100 to 0W/m\*\*2. Global mean = -47W/m\*\*2, Minimum = -144W/m\*\*2, Maximum = -4W/m\*\*2. (**Source:** NASA *Surface Radiation Budget Project*). *Average Net Longwave Radiation at the Earth's Surface: July 1983-1990* **(L\*)**
![](_page... | {
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The actual amount of net radiation being partitioned into each one of these components is a function of the following factors:
Presence or absence of water in liquid and solid forms at the surface.
*Specific heat* of the surface receiving the *net radiation*.
*Convective* and *conductive* characteristics of the r... | {
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#### **Annual Cycle of Air Temperature**
As the Earth revolves around the sun, locations on the surface may under go seasonal changes in air temperature because of annual variations in the intensity of *net radiation*. Variations in net radiation are primarily controlled by changes in the intensity and duration of ... | {
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Antarctica remains cold and below zero degrees Celsius due to the presence of permanent glacial ice which reflects much of the solar radiation received back to space.
**{**PRIVATE**}Figure 7m-2:** Mean January air temperature for the Earth's surface, 1959-1997. (**Source of Original Mod... | {
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The skier will of course move from the top of the hill to the bottom of the hill, with the speed of their descent controlled by the gradient or steepness of the slope. Likewise, wind speed is a function of the steepness or
gradient of atmospheric air pressure found between high and low pressure systems. When expresse... | {
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A wind that blows around curved isobars above the level of friction is called a *gradient wind*. Gradient winds are slightly more complex than geostrophic winds because they include the action of yet another physical force. This force is known as *centripetal force* and it is always directed toward the center of rotati... | {
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Like land/sea breezes, these wind systems are created by the temperature contrasts that exist between the surfaces of land and ocean. However, monsoons are different from land/sea breezes both spatially and temporally. Monsoons occur over distances of thousands of kilometers, and their two dominant patterns of wind flo... | {
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The intertropical convergence zone is identified on the **figures** by a red line. The formation of this band of low pressure is the result of solar heating and the convergence of the trade winds. In January, the intertropical convergence zone is found south of the equator (**Figure 7p-4**). During this time period, th... | {
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"source_pdf": "datasets/websources/Geography_v1/Geography/Fundamentals of Physical Geography By Michael ... |
Subsets and Splits
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