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Let us examine the phase information contained in the visibility function V(b). Consider the exponential in (33). Recall that the projected vector b is directed normal to the fringes and that $b/\lambda$ is the number of fringe cycles per radian (spatial frequency). Thus, $b \cdot \sigma/\lambda$ in (33) is the num... | {
"Header 1": "Phase of visibility function",
"token_count": 902,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The measured visibility $V(\mathbf{b})$ is thus a Fourier amplitude/phase that allows us to reconstruct the sky brightness $I(\sigma)$ . Those familiar with Fourier theory will recognize that (33) is the Fourier transform of the brightness distribution $I(\sigma)$ . It can be shown (not here) that the expression ca... | {
"Header 1": "Phase of visibility function",
"Header 2": "Sky brightness",
"token_count": 497,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The maps obtained with these techniques represent the content of the entire data set. Unfortunately, the artifacts invariably associated with incomplete *u*,v sampling cause the map to be less than perfect. For a given observation, one knows the artifact associated with a point source. Thus radio astronomers are able t... | {
"Header 1": "Phase of visibility function",
"Header 2": "Sky brightness",
"Header 3": "*Cleaning algorithms*",
"token_count": 834,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The utilization of multiple telescope spacings and orientations will improve image quality by removing artifacts. For our observation near the North Pole, the addition of different spacings obtained with additional telescopes would yield additional rings at different radii in the image, in effect flattening the region ... | {
"Header 1": "*Multiple baselines*",
"token_count": 277,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
There are currently a number of radio interferometric arrays in operation. There are several arrays containing at least eight telescopes extending over a few kilometers or more. Arrays of this size yield angular resolutions of down to 0.1 – 1. Examples are the Westerbork array in the Netherlands, the Very Large Array (... | {
"Header 1": "*Radio arrays*",
"token_count": 1187,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
It is possible to place telescopes on the earth's surface with spacings approaching the earth's diameter. Telescopes on different continents attain fringe spacings (and resolutions) less than 1 milliarcsecond (mas). While they provide exceptional angular resolution, only limited numbers of baselines can be attained, an... | {
"Header 1": "*Radio arrays*",
"Header 3": "Very long baseline interferometry (VLBI)",
"token_count": 493,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Optical "long baseline" interferometry is on the threshold of becoming a major tool in optical astronomy. A number of new generation telescopes (8-m class) are or have been built with multiple telescopes separated typically by $\sim 100$ m. Examples are the twin 10-m Keck telescopes in Hawaii and the four 8-m Very La... | {
"Header 1": "Optical and x-ray interferometry",
"token_count": 534,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Problem 7.21*. Verify the trigonometric conversions from (9) to (10) and from (17) to (18).
*Problem* 7.22. (a) The two telescopes of Fig. 2a on the earth's equator are separated E–W by 100 km and are tuned to a radio frequency of 1 GHz. They are observing a point source A on the celestial equator. What is the spati... | {
"Header 1": "Optical and x-ray interferometry",
"Header 3": "7.2 Two-telescope interference",
"token_count": 1047,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Problem 7.31. (a) Sketch qualitatively a few cycles of the response function R(t) (14) for a solitary source with no background for the following two cases. (i) a solitary point source and (ii) a source extended enough to modestly demodulate the response function (which remains sinusoidal; see next problem). Assume $\... | {
"Header 1": "Optical and x-ray interferometry",
"Header 3": "7.3 Mapping the sky",
"token_count": 1620,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Problem 7.41.* (a) The VLA array has 27 telescopes as shown in Fig. 9a. Nine are spaced along each of three arms of length 21 km arranged like three spokes of a wheel with equal angles (120◦) between the spokes. Consider an arrangement of the telescopes wherein the distances from the center on each arm scale as 1, 4, ... | {
"Header 1": "*7.4 Arrays of telescopes*",
"token_count": 338,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The flux of radiation arriving from a distant **point (unresolved) source** may be described with the **spectral flux density** *S*(ν, *t*) (W m−<sup>2</sup> Hz−1) which gives the flux as a function of frequency ν and time *t*. Integration of *S* over the frequency interval of the detector yields the **flux density** F... | {
"Header 1": "Point-like and extended sources",
"Header 3": "**What we learn in this chapter**",
"token_count": 542,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The quantities used to describe incoming radiation are presented in this chapter. These quantities should describe the rate of energy flow (power), its variation with time, its variation with frequency of the electromagnetic wave, and the direction(s) from which it flows. Note that "frequency" can also refer to oscilla... | {
"Header 1": "**8.1 Introduction**",
"token_count": 721,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The flux from an unresolved (point-like) object may be considered to be a parallel beam of light or a plane wave originating at "infinity". It impinges on the telescope with a given amount of energy deposited per second, per square meter, and per unit frequency interval at frequency ν (e.g., in the interval ν – 0.5 Hz ... | {
"Header 1": "*Spectral flux density*",
"token_count": 1431,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The total power flowing across unit area is called the *flux density* F (W/m2). It is obtained by integrating the spectral flux density *S*(ν) over the frequency band of interest,
➡ F ≡ <sup>ν</sup><sup>2</sup> ν1 *S*(ν) dν (W/m2; flux density) (8.6)
Again, this quantity is properly a vector since it is a flow that... | {
"Header 1": "*Flux density*",
"token_count": 246,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The luminosity *L* of a source is, in its usual meaning, the total power output (W) summed over *all* frequencies. In practice, one usually must specify the band of frequencies (of radiation) that are being measured, e.g., the visual *V* band, the entire optical band, or the 1–10 keV x-ray band. Normal stars like the s... | {
"Header 1": "*Luminosity*",
"token_count": 1305,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The traditional quantity used by optical astronomers to describe the intensity or brightness of a star has been the *visual magnitude m*. Hipparchus, the Greek astronomer and mathematician ( $\sim$ 135 BCE), with refinements by Ptolemy ( $\sim$ 150 CE), classified stars as having magnitudes 1, 2, 3, 4, 5, 6 where "1" w... | {
"Header 1": "8.3 Astronomical magnitudes",
"token_count": 525,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A typical detector in the IR–optical–UV region has a response that is proportional to the *number* of photons impinging upon it in a given time. A quantitative measure of this is the *photon spectral flux density S*p(ν) which is the number of photons (rather than their energy) at frequency ν per unit frequency, area, a... | {
"Header 1": "*Magnitudes and fluxes*",
"token_count": 1071,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The magnitude of a star can be measured in any of a number of wavelength bands. Thus when specifying a magnitude, the spectral band that was measured is usually specified. The "3-color" system mentioned above is the traditional system often used, namely the *UBV* system: ultraviolet, blue, visual (yellow). Each spectra... | {
"Header 1": "*Spectral color bands*",
"token_count": 1379,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Since magnitudes are not physical (SI) units, it is often necessary to convert magnitudes to (spectral) flux densities S (W m<sup>-2</sup> Hz<sup>-1</sup>). The flux densities $S_0$ that correspond to the magnitude zero for any color are given in the rightmost column
$<sup>^</sup>b$ 1.0 nm (nanometer) = $1.0 \tim... | {
"Header 1": "Conversion from magnitudes to SI units",
"token_count": 1952,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The luminosity of a celestial body can be cast into the system of magnitudes. The absolute magnitude M is a direct measure of luminosity, the power (W) emitted by the star in the specified spectral band. It is defined as the magnitude of a star if it were at the standard distance of 10 parsecs. (1 pc $\approx$ 3.26 L... | {
"Header 1": "Conversion from magnitudes to SI units",
"Header 3": "Absolute magnitudes – luminosity",
"token_count": 1158,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The bolometric magnitude $m_{bol}$ was defined above (see Table 1). It is based upon the total energy flux in and near the optical band (IR, optical, UV). The distribution of radiation from a star approximates a blackbody spectrum with superimposed absorption lines; it peaks in the visible range for many stars and fa... | {
"Header 1": "Conversion from magnitudes to SI units",
"Header 2": "Bolometric magnitude",
"token_count": 799,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The *absolute* bolometric magnitude is a direct measure of the *bolometric* luminosity of a star, i.e., the energy output integrated over the IR, visual, and UV, as appropriate for the temperature of the star. The bolometric magnitude $M_{\rm bol}$ of an arbitrary star is related, by definition, to its luminosity L a... | {
"Header 1": "Absolute bolometric magnitude and luminosity",
"token_count": 839,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A hypothetical antenna with a narrow beam and narrow bandwidth could survey the entire sky and produce a map of the brightness of the sky. Your eye does this when it surveys the sky and so does the radio antenna that produces radio sky maps. Since different antennae have different areas, beam sizes, and bandwidths, it ... | {
"Header 1": "*Concept of specific intensity*",
"token_count": 1179,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
If the specific intensity of the sky is *not* uniform over the received band of frequencies and beam angles, one must sum the contributions by each element of the sky to obtain the received power. In other words, one must integrate the specific intensity $I(\theta,\phi,\nu)$ over the observed angles and frequencies, ... | {
"Header 1": "*Concept of specific intensity*",
"Header 3": "Power received by antenna",
"token_count": 890,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The specific intensity can be measured precisely only with an *ideal* narrow-beam and narrow-bandwidth antenna. The ideal measure of $I(\theta, \phi, \nu)$ with perfect angular and frequency resolution can not be attained from measurements with a real antenna. Our knowledge of the sky can be no better than the resolu... | {
"Header 1": "*Concept of specific intensity*",
"Header 3": "Average specific intensity",
"token_count": 765,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Relation to specific intensity
The spectral flux density S(v) used for the study of point sources can be expressed in terms of the specific intensity. Consider the flux from an extended celestial source impinging upon a flat surface of unit area (a very simple antenna) and calculate the amount of flux passing through... | {
"Header 1": "*Concept of specific intensity*",
"Header 3": "Spectral flux density revisited",
"token_count": 619,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
If a source is point-like at celestial position θ ,φ we mean only that its angular size is smaller than the resolution of the instrument imaging it. In this case one measures the flux density *S*. What can one say about the specific intensity? If the resolution (beam size) is Ωbeam, one only knows that the solid angle ... | {
"Header 1": "*Specific intensity of pulsars*",
"token_count": 956,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Heretofore, the specific intensity has been used as a standard way to describe the power received by a telescope. However, the units of specific intensity, W m<sup>-2</sup> Hz<sup>-1</sup> sr<sup>-1</sup>, are also appropriate for the description of the power *emitted from* a surface. Consider a mathematical surface (F... | {
"Header 1": "*Specific intensity of pulsars*",
"Header 3": "Power emitted from a surface",
"token_count": 2049,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The origin of this equality lies in a fundamental theorem of physics, known as Liouville's theorem. It makes use of the concept of *phase space*, a six-dimensional space with three spatial dimensions, *x*, *y*,*z* and three momentum dimensions, *px* , *py* , *pz*. The density in phase space is the number of particles o... | {
"Header 1": "*Liouville's theorem*",
"token_count": 452,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The names and symbols for the specific intensity and related quantities are hardly standardized in practical use. This is less than helpful to the suffering student (and teacher). However, we try to be consistent within this text. A summary of the names and units of the several energy-related quantities we use are give... | {
"Header 1": "*Energy flow – names, symbols, and units*",
"token_count": 655,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A source that appears extended on the sky in the angular directions θ ,φ is undoubtedly also extended along the line of sight, toward and away from the observer. Such diffuse objects are three-dimensional clouds of material, often gases with some degree of ionization. It is convenient and physical to consider the amoun... | {
"Header 1": "*Volume emissivity*",
"token_count": 290,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Let the emission from a volume element in the cloud be isotropic; the radiation leaves the source in all directions equally. Since all directions constitute $4\pi$ sr, the power emitted into one steradian is
Power into 1 sr =
$$\frac{j}{4\pi}$$
(W m<sup>-3</sup> Hz<sup>-1</sup> sr<sup>-1</sup>) (8.46)
Further ass... | {
"Header 1": "*Volume emissivity*",
"Header 3": "Relation to specific intensity",
"token_count": 1797,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Problem 8.21. (a) What is the (partial) luminosity within a narrow $10^4$ Hz bandwidth at frequency $\nu$ of a radio source radiating isotropically if it is detected at distance 3000 LY with a spectral flux density S=1.0 Jy in this band? Repeat for the broad band 50 to 200 MHz if $S(\nu)$ is constant at 1.0 Jy ov... | {
"Header 1": "*Volume emissivity*",
"Header 3": "8.2 Unresolved point-like sources",
"token_count": 995,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Problem 8.41*. When the moon is full, its magnitude (summed over its entire surface) is V = -12.7. The diameter of the moon is about 30'. (a) The resolution of the human eye is $\sim 1'$ . What is the magnitude of the moon per square arcminute? (b) What is the magnitude per square arcsecond; the resolution of a groun... | {
"Header 1": "*Volume emissivity*",
"Header 3": "8.4 Resolved \"diffuse\" sources",
"token_count": 1329,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The **information content** in the radiation recorded in observations allows astronomers to derive the **properties** of celestial objects. The **ranges of the values** of these properties are found to be "astronomically" large. **Luminosities** are derived from measured fluxes and distances. The solar luminosity, 3.8 ... | {
"Header 1": "**What we learn in this chapter**",
"token_count": 635,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The distance to a celestial object is often crucial to determining its size or luminosity. Unlike the angular position of a star or galaxy on the celestial sphere, distances often can be obtained only indirectly. A significant fraction of astronomical history has been dedicated to the determination of distances. Even n... | {
"Header 1": "**9.1 Introduction**",
"token_count": 408,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The luminosity of an object as discussed in Section 8.2 is the energy it emits per unit time, e.g., joules per second (J s−1) or the equivalent unit, watt (W). It can refer to the energy emitted over the entire electromagnetic spectrum, but it can mean the energy emitted in a more restricted frequency band, e.g., the v... | {
"Header 1": "**9.2 Luminosities**",
"token_count": 1628,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The masses of the sun and earth may be determined from their mutual motions about their center of mass (*barycenter*). In general for elliptical orbits, the analysis is relatively involved. But if the orbits are circular and if one of the two objects is much more massive than the other, *M m*,it is quite straightforwar... | {
"Header 1": "**9.2 Luminosities**",
"Header 3": "*Earth and sun*",
"token_count": 802,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The mass of the (MW) Galaxy and the masses of other spiral galaxies may be estimated in a manner similar to that used for the earth and sun. Stars in a spiral galaxy rotate in a more or less organized manner about the center at some radius with a corresponding period of rotation. Individual stars can not be resolved ex... | {
"Header 1": "**9.2 Luminosities**",
"Header 3": "Spiral galaxies and the Galaxy",
"token_count": 955,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The only objects with substantially greater masses than galaxies, other than the universe itself, are collections of galaxies, known as *clusters of galaxies*. The nearby (~55 MLY) Virgo cluster contains about 2500 galaxies. The galaxies move about under the influence of all the other galaxies in the cluster; think of ... | {
"Header 1": "**9.2 Luminosities**",
"Header 3": "Clusters of galaxies and the virial theorem",
"token_count": 718,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Temperature is one of the most basic quantities in physics and astrophysics. At its simplest, according to the approximation *h*ν ≈ *kT* (2.15), the frequencies of detected photons may be an indicator of the temperature of the originating bodies. For example, photons from the sun at optical frequencies tell us that the... | {
"Header 1": "*Thermal and non-thermal radiation*",
"token_count": 365,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The color temperature *T*<sup>c</sup> is derived from broadband *UBV* photometry, the measurement of fluxes in the *U, B,* and *V* bands, under the assumption that the observed object has a blackbody spectrum. The relative flux densities in these bands are a measure of the color of the object (Section 8.3). The nature ... | {
"Header 1": "*Color temperature*",
"token_count": 427,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The distribution of light from normal stars roughly approximates the continuum spectrum from a blackbody wherein each square meter emits a flux of σ *T* <sup>4</sup> (W/m2) and <sup>σ</sup> is the Stefan–Boltzmann constant, <sup>σ</sup> <sup>=</sup> <sup>5</sup>.<sup>67</sup> <sup>×</sup> <sup>10</sup>−<sup>8</sup> Wm−... | {
"Header 1": "*Effective temperature*",
"token_count": 383,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The *excitation temperature* is that deduced from the populations of atomic excited states observed in stellar spectra. The higher the energy of the occupied states, the higher the temperature. For example, the greater the number of hydrogen atoms in the *n* = 2 state relative to the number in the *n* = 1 state, the hi... | {
"Header 1": "*Effective temperature*",
"Header 3": "*Excitation temperature*",
"token_count": 781,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The *ionization temperature* is also determined through spectral studies. It depends upon the extent to which different ionization states (as distinct from the excitation states just discussed) appear in the spectrum, e.g., the ratio of He I to He II, neutral to ionized helium. The relative numbers of ions in the sever... | {
"Header 1": "*Effective temperature*",
"Header 3": "Ionization temperature",
"token_count": 357,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The Saha equation allows one to obtain the ionization temperature from measurements of the relative numbers of atoms of a given species that are in adjacent ionization states (Figure 1). The equation follows from consideration of *detailed balance* wherein the probabilities of absorption and emission of an electron by ... | {
"Header 1": "*Saha equation*",
"token_count": 2043,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The angular position of a star on the celestial sphere requires only the two angular coordinates. Indirect means must be used to obtain the
<sup>&</sup>lt;sup>b</sup> The moon distance varies by 11% due to its elliptical orbit.
<sup>&</sup>lt;sup>c</sup> Mean value = semimajor axis of Earth orbit; the earth–sun dis... | {
"Header 1": "*Saha equation*",
"token_count": 676,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The distance to the moon from the earth center was determined with poor accuracy by the Greek astronomer Aristarchus (∼270 BCE) from timing of lunar eclipses. This method was improved upon by a later Greek astronomer, Hipparchus ( $\sim$ 140 BCE). Ptolemy (second century CE) used a different method; he measured the par... | {
"Header 1": "*Moon, earth, and sun*",
"token_count": 1048,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The technique of trigonometric parallax yields the distances of the closer stars. As described in Section 4.3, nearby stars appear to move relative to the more distant stars as the earth moves about the sun (see Fig. 4.2). This motion appears as a circle, ellipse, or line depending on the star's location relative to th... | {
"Header 1": "*Moon, earth, and sun*",
"Header 3": "Trigonometric parallax",
"token_count": 684,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The motion of the cluster as a whole typically turns out to be much greater than the motions of individual stars within the cluster. Thus one can visualize a cluster of individual stars moving through space as a group, each with the velocity of the cluster. If the radial portion of the motion is such that the cluster i... | {
"Header 1": "*Moon, earth, and sun*",
"Header 2": "Convergence",
"token_count": 1347,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
In traditional parallax determinations of distances, the baseline motion of the earth is limited to ±1 AU. The annual oscillatory motion of the earth leads to an oscillatory motion of the apparent track of the star under study. It is thus easy to identify unambiguously the apparent stellar motion (parallax) that is due... | {
"Header 1": "*Secular parallax*",
"token_count": 1505,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The method of*statistical parallaxes* makes use of the other component of the proper motion in the plane of the sky, the *tau component* (dθ/d*t*)<sup>τ</sup> (Fig. 4). This motion is perpendicular to the solar motion and is unaffected by it. Only the peculiar motion of the star affects the tau component, and this, by ... | {
"Header 1": "*Statistical parallax*",
"token_count": 698,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Subsequent steps in the distance ladder involve the identification of *standard candles*. The calibration of the standard candles is a step-by-step process. First the distance to a nearby object is obtained, say by trigonometric parallax. This distance, together with the measured flux, yields the luminosity *L* accordi... | {
"Header 1": "*Statistical parallax*",
"Header 3": "*Standard candles*",
"token_count": 346,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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All normal stars can be classified according to their spectral types. Each type is characterized by a particular luminosity and a unique combination of spectral features that follow from the mass of the star and its stage of evolution. Among the stars burning hydrogen in their core (like our sun), the low-mass stars ar... | {
"Header 1": "*Statistical parallax*",
"Header 3": "Spectroscopic classification",
"token_count": 525,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The distance to the center of our galaxy has been obtained by radio astronomers with a method that made use of a cluster of H2O masers in a star-forming region in the close vicinity of the galactic center (Fig. 5). The masers are probably condensations of interstellar gas and dust in the vicinity of a luminous newly fo... | {
"Header 1": "*Statistical parallax*",
"Header 3": "*Galactic center and Crab nebula distances*",
"token_count": 717,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Cepheid variables are luminous stars that are 300 to 30 000 times the luminosity of the sun (300–30 000 $L_{\odot}$ ). Their luminosities vary periodically with periods of days (1–100 d); see Fig. 6a. The luminosity variation is due to oscillation in the size (height) and temperature of the stellar atmosphere associat... | {
"Header 1": "*Statistical parallax*",
"Header 3": "Cepheid and RR Lyrae variables",
"token_count": 1333,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The redshift may be interpreted as due to the Doppler effect (16) brought about by high recessional velocities; the emitting galaxies are moving rapidly away from the (MW) Galaxy. In this interpretation, Hubble's result tells us that the galaxy recessional speed v is proportional to its distance *r* from the Galaxy. Th... | {
"Header 1": "*Receding galaxies*",
"token_count": 322,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
In the simplest picture of an expanding universe, the speed of the individual galaxies do not change as they fly away from each other. Think of the galaxies as raisins in a bread being baked in an oven. As the bread expands (linearly with time), the raisins move farther and farther from one another. The recession speed... | {
"Header 1": "*Receding galaxies*",
"Header 3": "*Expanding universe*",
"token_count": 401,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The value of $H_0$ is obtained from measurements of v and r for a large number of galaxies; it depends upon careful distance calibrations. This value has been and is still a much discussed number. Proponents have argued for values ranging from 15 to 30 km s<sup>-1</sup> MLY<sup>-1</sup> (or, multiplying by 3.26 LY/pc... | {
"Header 1": "*Receding galaxies*",
"Header 3": "Value of Hubble constant",
"token_count": 1257,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Returning to the simplest model wherein the expansion speed of any given galaxy is constant in time, one can find an approximate "size" of the "observable" universe. We presume this distance to be that where the expansion speed is approaching the speed of light. At this distance, one could not observe a celestial objec... | {
"Header 1": "*Receding galaxies*",
"Header 3": "Size and age of the universe",
"token_count": 450,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The quest for standard candles as independent indicators of distance continues to this day. Modern sensitive instrumentation and the ingenuity of contemporary astronomers are bringing about important advances. The objects now used as standard candles include supernovae at their peak brightness, galaxies of the spiral a... | {
"Header 1": "Extragalactic \"standard candles\"",
"token_count": 271,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A hypothetical example of the use of global properties as a distance indicator would be to determine the average luminosity of all the globular clusters in a distant galaxy. Globular clusters are compact collections of 105 – 10<sup>6</sup> stars of which several hundred surround the center of the Galaxy (Figs. 1.9 and ... | {
"Header 1": "*Luminosity functions*",
"token_count": 653,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Galaxies themselves have been used as standard candles. It turns out that large *spiral galaxies* are more luminous if they are more massive. This is not surprising because a greater mass indicates more stars. Also, the more massive galaxies cause the gas in their outer regions to rotate around their centers faster tha... | {
"Header 1": "*Line-broadening in galaxies*",
"token_count": 447,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A very different method invokes pixel to pixel fluctuations of intensity in the images of elliptical galaxies. A distant galaxy will generally appear as a diffuse blob due to the thousands of unresolved stars in each image element, that is, in each pixel of a CCD. (Charge-coupled devices are described in Section 6.3.) ... | {
"Header 1": "*Line-broadening in galaxies*",
"Header 3": "*Surface brightness fluctuations*",
"token_count": 1966,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
$\sigma_b$ Flux $\sigma_b^2/b$ (41) $^b$ | $m$ $f$ $fm$ $f\sqrt{m}$ $\frac{(f\sqrt{m})^2}{fm} = f$ | $ 4m f/4 (f/4) 4m = fm (same) \frac{1}{4} f \sqrt{4m} = \frac{1}{2} f \sqrt{m} (\frac{1}{2} f \sqrt{m})^2 (fm)^{-1} = f/4 $ |
Table 9.3. Surface brightness fluctuations, example<sup>a</sup>
The illustration of F... | {
"Header 1": "*Line-broadening in galaxies*",
"Header 3": "*Surface brightness fluctuations*",
"token_count": 2031,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
(It is impossible to extract more energy than this.) Twice your answer will be the famed Schwarzschild radius, the event horizon for a non-rotating black hole. What is the Schwarzschild radius for a quasar mass of $10^8~M_{\odot}$ ? What is it for the sun $(1~M_{\odot})$ ? (c) Write in terms of *m* an expression for ... | {
"Header 1": "*Line-broadening in galaxies*",
"Header 3": "*Surface brightness fluctuations*",
"token_count": 2036,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Our knowledge of celestial objects must take into account **absorption** and **scattering** of photons as they travel to earth observers. These processes are highly **frequency dependent** and thus affect some bands more than others. Photon–electron interactions include **Rayleigh**, **Thomson** and **Compton scatterin... | {
"Header 1": "Absorption and scattering of photons",
"Header 3": "**What we learn in this chapter**",
"token_count": 412,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Many photons en route to the earth from distant objects do not arrive safely. They may encounter an electron, an atom, a molecule, a dust grain, a star or planet, or *10.1 Introduction* 299
even gravitational fields and be absorbed or scattered into different directions of propagation. Scattered photons may appear to... | {
"Header 1": "Absorption and scattering of photons",
"Header 3": "**10.1 Introduction**",
"token_count": 885,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A relatively low-energy photon may be absorbed and immediately re-emitted by an atom without causing the atom or molecule to change its energy state. This is known as *Rayleigh scattering* which may be understood classically as the interaction of the photon and one of the electrons. Consider the electron to be bound by... | {
"Header 1": "Absorption and scattering of photons",
"Header 2": "Rayleigh scattering",
"token_count": 1227,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
If the photon energy is very high (in this context), approaching the rest-mass energy of an electron, *<sup>h</sup>*<sup>ν</sup> <sup>≈</sup> *<sup>m</sup>*e*c*<sup>2</sup> <sup>=</sup> 511 keV, the probability of scattering from free electrons begins to decrease rapidly with frequency. This high-energy extension of Th... | {
"Header 1": "*Compton scattering*",
"token_count": 500,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Gamma rays at extremely high energy ( $>10^{14}$ eV) will interact with the cosmic microwave background (CMB) radiation and thereby cease to exist. The interaction converts two photons into an electron–positron pair, sometimes simply called an "electron pair",
$$\gamma_{\text{gamma}} + \gamma_{\text{cmb}} = e^{-} + ... | {
"Header 1": "*Compton scattering*",
"Header 3": "Photon absorption in the CMB",
"token_count": 707,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Photoelectric absorption and absorption lines*
An important example of the attenuation of a photon beam in astrophysics is absorption by the hydrogen atoms in interstellar space. In such an interaction, the photon is completely absorbed by the atom and ceases to exist. The converted photon energy can excite the atom... | {
"Header 1": "*Photon–atom interactions*",
"token_count": 323,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
When a photon is absorbed by an electron or atom (photoelectric absorption), a portion of the energy may be re-emitted by the atom in the form of spectral lines that can be measured from the earth. Dramatic examples of this are the *H II regions* in the vicinity of newly formed and very hot stars. Energetic ultraviolet... | {
"Header 1": "*Photon–atom interactions*",
"Header 3": "*Emission nebulae*",
"token_count": 602,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Photons of energies $\gtrsim 10 \, \text{MeV}$ (gamma rays) that pass close to an atomic nucleus will interact with the electric field and spontaneously convert to an electron–positron pair. This is known as *pair production*. The process is fundamentally the same process as that discussed above (3); the electric fie... | {
"Header 1": "*Photon–atom interactions*",
"Header 3": "Pair production near a nucleus",
"token_count": 211,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Optical photons traveling in the plane of the Galaxy are heavily scattered by clouds of dust (*interstellar grains*) which limit the range of visibility to, very roughly, $5000 \, \text{LY}$ , a distance that varies considerably for different viewing directions. (The Galaxy is $\sim 100\,000 \, \text{LY}$ in diamete... | {
"Header 1": "*Photon–atom interactions*",
"Header 3": "10.3 Extinction of starlight",
"token_count": 1219,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Extinction coefficient*
The *extinction AV* is the number of magnitudes by which light in the visual *V* band (Section 8.3) is dimmed by the intervening dust. The average or typical amount of extinction of visible light (*V* band) in the plane of the Galaxy (within ∼500 LY of the plane) is about 0.6 magnitudes for e... | {
"Header 1": "*Extinction parameters*",
"token_count": 489,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
It is fortunate that this extinction is sufficiently low to let us see out of the Galaxy in directions away from the galactic plane. In this way we can study and learn about other galaxies. In the direction of the galactic pole, perpendicular to the plane of the Galaxy, the visual-band extinction is about 0.2 magnitude... | {
"Header 1": "*Extinction parameters*",
"Header 3": "Extragalactic sources",
"token_count": 1348,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The measured dependence of the extinction $A(\nu)$ upon frequency over a wide range of frequency is given in Fig. 3. The extinction is due to the integrated effect of grains of many different sizes, each with its own efficiency for scattering the light. The ordinate is the ratio of $A(\nu)/A_V$ . The effective frequ... | {
"Header 1": "*Extinction parameters*",
"Header 3": "Frequency dependence",
"token_count": 518,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The grains in the Galaxy are quite closely clumped with the hydrogen in the interstellar medium. We now describe how this was determined.
Studies from satellites of individual hot stars at ultraviolet frequencies provide information about the interstellar hydrogen. The ultraviolet photons en route to the earth from t... | {
"Header 1": "*Extinction parameters*",
"Header 3": "Dust-hydrogen association",
"token_count": 1449,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The concept of cross section is illustrated in Fig. 5a. The atom is visualized as having a target area, or "bull's-eye", which a photon may or may not strike. This target area is called the *cross section* $\sigma$ with units of m<sup>2</sup>. The value of the cross section for a given atom is tiny, e.g., $10^{-24}$... | {
"Header 1": "*Extinction parameters*",
"Header 3": "Cross section as a target",
"token_count": 868,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A beam of photons impinges on a thin slab of material (Fig. 5b). The slab has thickness dx and area A ( $m^2$ ). It contains many scattering atoms, each with cross section $\sigma$ ( $m^2$ ) and a random position in the slab. The slab is taken to be sufficiently thin so that the sum of all the individual cross sectio... | {
"Header 1": "*Extinction parameters*",
"Header 3": "Exponential absorption",
"token_count": 616,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A calculation of the average propagation distance, known as the *mean free path* $x_m$ , turns out, fortuitously, to be equal to the 1/e distance. Thus,
$$x_{\rm m} = (\sigma n)^{-1}$$
(m; mean free path) (10.21)
and (20) may be written,
$$N(x) = N_i \exp(-x/x_{\rm m})$$
(10.22)
The intensity N(x) decreases to... | {
"Header 1": "*Extinction parameters*",
"Header 3": "Mean free path",
"token_count": 1259,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Astronomers use yet another form of notation to describe absorption, the *optical* depth, $\tau \equiv x/x_{\rm m} = \kappa \rho x$ , which is simply the thickness in units of the mean free path, a dimensionless quantity. The formal definition, allowing for variation of $\kappa$
10.4 Cross sections 319
| Thicknes... | {
"Header 1": "Optical depth",
"token_count": 644,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Since a cross section $\sigma_g$ gives rise to extinction of starlight by interstellar grains, one can develop a relation between cross section $\sigma_g$ and the extinction coefficient $A_V$ . It turns out that the two quantities are proportional to one another as we now demonstrate.
Consider starlight traversi... | {
"Header 1": "Cross section and extinction coefficient",
"token_count": 905,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The photoelectric effect occurs when a photon ejects an electron from an atom; the photon ceases to exist. The most abundant atom in interstellar space, hydrogen, can be ionized by photons with energies above $13.6 \,\mathrm{eV}$ ( $\lambda = 91.2 \,\mathrm{nm}$ ), i.e., by ultraviolet or x-ray photons. For pure hydr... | {
"Header 1": "Cross section and extinction coefficient",
"Header 3": "Photoelectric effect",
"token_count": 1136,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The relative abundances of the most abundant elements in the solar system are presented in Table 2. They come from studies of meteorites, stony solar-system objects that have fallen to the earth's surface from their circumsolar orbits. Studies of the spectra of the sun's photosphere yield abundances that are in good ag... | {
"Header 1": "Cross section and extinction coefficient",
"Header 3": "Abundances by number",
"token_count": 212,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
In contrast to relative *number* abundances, one can describe the relative abundances in terms of their contributions to the *mass* of the interstellar medium. The fraction of
<sup>&</sup>lt;sup>a</sup> Abundances from meteorites; adapted from N. Grevesse and E. Anders, in *Cosmic Abundances of Matter*, ed. C. J. Wad... | {
"Header 1": "Mass fractions",
"token_count": 608,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
An effective cross section in the interstellar medium may be calculated from our knowledge of the relative abundances of the elements (Table 2). The result for the UV to x-ray region is plotted in Fig. 7a. (The optical region at ∼2 eV is off the plot,
Figure 10.7. Effective photoelectri... | {
"Header 1": "*Effective cross section*",
"token_count": 2008,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The problem can be turned around to find the photon energy at which the photon flux will be attenuated to $e^{-1} = 0.368$ of its initial value as it travels from the galactic center to the sun. Adopt the values of $n_{\rm H}$ and r in (45), and $\mathscr{F}_{\rm p}/\mathscr{F}_{\rm p,0} = 0.368$ , and then solv... | {
"Header 1": "*Effective cross section*",
"token_count": 644,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Similar calculations provide the entire 37% curve as well as the $e^{-0.25} = 0.78 = 78\%$ and $e^{-4} = 1.8\%$ curves (Fig. 8). At a given $N_{\rm H}$ , higher x-ray energies yield greater survival fractions because the cross sections are lower, while lower photon energies yield smaller survival fractions. The en... | {
"Header 1": "Astronomy through gases",
"token_count": 1131,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Problem 10.31*. (a) Calculate the mass $m_{\rm g}$ of a typical interstellar grain. Adopt the density of water ( $\rho_{\rm G}=1000~{\rm kg/m^3}$ ) for the grain density and a spherical grain size (diameter) of 0.2 μm. Compare to the mass and size of a hydrogen atom. (b) Find the average *number* density (#/m³) of g... | {
"Header 1": "Astronomy through gases",
"Header 3": "10.3 Extinction of starlight",
"token_count": 1602,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The distribution with frequency of radiation from a source is called a **spectrum**. It can be plotted as an energy spectrum or as a number spectrum and as a function of either frequency or wavelength. **Conversions** from one to another are possible and useful. **Continuum spectra** are without spectral lines though s... | {
"Header 1": "**What we learn in this chapter**",
"token_count": 393,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The spectral distribution of electromagnetic radiation from a celestial object can reveal much about the physical processes taking place at the object. The *spectral*
Figure 11.1. Idealized spectra of radiation: sketches of (a) line emission, (b) line absorption, (c) a broadened emissio... | {
"Header 1": "**What we learn in this chapter**",
"Header 3": "**11.1 Introduction**",
"token_count": 375,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The (*energy*) specific intensity has previously been defined (8.26) as follows:
$$I(v)$$
: (W m<sup>-2</sup> Hz<sup>-1</sup> sr<sup>-1</sup>) (Energy specific intensity) (11.1)
This quantity is appropriate for a diffuse (resolved) celestial source, one larger in angular size than the telescope resolution. It is of... | {
"Header 1": "**11.2 Plots of spectra**",
"Header 3": "*Energy and number spectra*",
"token_count": 1122,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The conversion of the specific intensity between the frequency and wavelength units was quoted in (8.43). We now derive this conversion, but we do it for the flux density *S* (W m−<sup>2</sup> Hz−1; frequency units). The flux density in wavelength units *S*<sup>λ</sup> is power per unit area per unit wavelength interva... | {
"Header 1": "*Frequency and wavelength*",
"token_count": 800,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Energy units in the reference band in use by high-energy astronomers are usually keV or MeV rather than joules. Thus one might see S (erg cm<sup>-2</sup> s<sup>-1</sup> keV<sup>-1</sup>). The conversion of this expression to our standard SI units S (W m<sup>-2</sup> Hz<sup>-1</sup>) requires only numerical multiplicati... | {
"Header 1": "*Frequency and wavelength*",
"Header 3": "Frequency and energy",
"token_count": 375,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Data may be taken with various frequency resolutions. A high resolution (narrow bandwidth) allows one to detect and study narrow spectral lines. Since a number must be stored or telemetered for each of the many narrow spectral bands, a high resolution can require a lot of data storage space (or telemetry from a space v... | {
"Header 1": "Spectral bin widths",
"token_count": 362,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
On the atomic level, continuum spectra can arise from *free–bound* or from *bound– free* transitions of the atom. The former occurs when an electron makes a transition from an unbound (free) state to a bound state, thereby emitting a photon with the energy of the transition. Since the unbound states are a continuum of ... | {
"Header 1": "*Free–bound, bound–free and free–free transitions*",
"token_count": 450,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
In a plasma cloud, the radiation arises when electrons are accelerated in near collisions with ions and thereby emit photons. The term "bremsstrahlung" means "braking radiation" in German. In an electron–ion near collision, the less-massive electron undergoes a large acceleration due to the mutual Coulomb force. An acc... | {
"Header 1": "*Radiation from a hot plasma*",
"token_count": 524,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The quantity that one can measure from afar is the specific intensity I which is related to j according to (8.50),
$$I = \frac{1}{4\pi} \int_0^A j(r) dr$$
(Specific intensity; hydrogen plasma; W m<sup>-2</sup> Hz<sup>-1</sup> sr<sup>-1</sup>)
$$\propto \frac{1}{4\pi} \int_0^A n^2 T^{-1/2} \exp(-h\nu/kT) dr$$
where ... | {
"Header 1": "*Radiation from a hot plasma*",
"Header 3": "Plasma parameters determined",
"token_count": 1286,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
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