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*A*s noted earlier (Section 3.2), parallax is due to the changing position of the earth as it moves about the sun. A star relatively close to the solar system will appear to move relative to the background stars (Fig. 2). Parallax is the same phenomenon that can be observed while riding in an automobile; the foreground... | {
"Header 1": "*Sun and the ecliptic*",
"Header 3": "*Parallax of star positions*",
"token_count": 1108,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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There is also a small shift of star positions that arises from the *velocity* of the earth as it orbits the sun. (Parallax is a consequence of the varying *position* of the earth.) This effect is called *stellar aberration*. It should be calculated with special relativity, but since the earth is traveling at a speed mu... | {
"Header 1": "*Sun and the ecliptic*",
"Header 3": "*Stellar aberration*",
"token_count": 714,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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In the case of the earth, the torque is due to the gravitational attraction of the sun, moon, and planets acting on the earth. These bodies could not exert a torque on the earth if it were perfectly spherical. However, the earth has an equatorial bulge; its radius is larger at the equator than at the poles. The sun, mo... | {
"Header 1": "Torque due to a ring of mass",
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What happens to the calendar and the seasons as the earth precesses? Consider a future date 12 885 yr from now. The sun will pass annually through Pisces as it always does, but as noted above it will be passing from north to south in earth coordinates. It thus will be the beginning of fall in the northern hemisphere; w... | {
"Header 1": "*Calendar*",
"token_count": 895,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Proper motion is a rate of change of angular position, $\mu = \mathrm{d}\theta/\mathrm{d}t$ (radians/s) on the celestial sphere, relative to the distant galaxies. It is often given in the mixed units of milliarcsec per year (mas/yr) by astronomers. The direction of movement is given by specifying the rate of movement... | {
"Header 1": "*Calendar*",
"Header 3": "Motion on celestial sphere",
"token_count": 776,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The velocity of a star in three-dimensional space relative to the barycenter of its neighbors is called its *peculiar motion*. It is likely due to gravitational interactions with other stars at some time in the past. Stars that approach close to one another experience mutual gravitational attraction and hence accelerat... | {
"Header 1": "*Calendar*",
"Header 2": "Peculiar motion and local standard of rest",
"token_count": 386,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The moon orbits the earth every 27.32 d (the *tropical month*, the time it takes between south-to-north crossings of the celestial equator, i.e., equinox to equinox. The sidereal period (relative to the stars) is nearly identical to this, within 10−<sup>4</sup> d. Thus the moon changes its position on the celestial sph... | {
"Header 1": "*\"Orbits\" of the moon and sun*",
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A solar eclipse requires that both the sun and the moon arrive at the node at nearly the same time. This coincidence need not be precise because (*i*) the two orbits are tilted only 5◦ from one another, and (*ii*) the observer can be positioned anywhere north or south on the earth to bring the sun and moon into suffici... | {
"Header 1": "*Total and partial solar eclipses*",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Since ancient times, it has been noted that a lunar eclipse will always be followed by a similar eclipse 6585.3 d later (18 yr 11.3 d if there are 4 leap years), and later it became known that solar eclipses also recurred with this same interval. Any solar eclipse will surely be followed 6585.3 d later by another of si... | {
"Header 1": "*Total and partial solar eclipses*",
"Header 3": "*The 18-year saros cycle*",
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Total eclipses are a wondrous experience. During the eclipse, one can see with naked eye the solar corona extending several solar diameters beyond the (covered) solar disk. Beautiful red prominences can sometimes be seen with the naked eye. These are giant loops of hot gas emitting the red Balmer line of hydrogen. The ... | {
"Header 1": "*Wonder and science*",
"token_count": 534,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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An x-ray photo of the sun was obtained during the 1991 July 11 eclipse from a rocket launched at White Sands, New Mexico (USA) exactly when the sun was totally eclipsed in Hawaii (Fig. 6a). At this time in New Mexico, the (partial) eclipse had not quite started. The looming and approaching shadow of the moon is seen to... | {
"Header 1": "*Corona in x rays and visible light*",
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Fifteen days before or after a solar eclipse, the moon has moved to the opposite side of the earth, almost directly opposite the sun. Since the earth–moon–sun alignment was quite precise just 15 d earlier, it is not unlikely that the moon will now enter into the shadow cast by the earth. This is known as a *lunar eclip... | {
"Header 1": "*Lunar eclipses*",
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The motions of the sun and moon on the celestial sphere are both steadily eastward. In contrast the motions of the planets on the celestial sphere are quite complex. The planets all orbit the sun in the same direction as the earth (eastward viewed from the sun), but, viewed from the earth, they will move both eastward ... | {
"Header 1": "*Lunar eclipses*",
"Header 3": "*Planets*",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The study of celestial motions requires a definition of "time". It turns out that time keeping is not a simple process. There are many factors that contribute to this complexity. It was difficult in early times when clocks at different geographic locations could not be easily synchronized, and it is difficult in modern... | {
"Header 1": "**4.5 Measures of time**",
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The most fundamental time keeping is based on the motions of the sun and stars as they pass overhead with seasonal variations.
#### Sidereal time
As the earth rotates under the sky, the zenith of a given observatory (e.g., Palomar Mountain) moves along the celestial sphere from west to east (Fig. 3.1). Thus, as we ... | {
"Header 1": "**4.5 Measures of time**",
"Header 3": "*Time according to the stars and sun*",
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As stated above, one can set a UT clock by observing the stars passing over Greenwich at midnight on Jan. 1. When the appropriate meridian of the celestial sphere (sidereal time $\sim$ 6 h 42 m = 6.7 h on Jan. 1) transits the zenith, it is exactly 0 h UT. The current relation used to set UT in terms of the star transi... | {
"Header 1": "**4.5 Measures of time**",
"Header 3": "Greenwich mean sidereal time (GMST) at 0 h UT",
"token_count": 712,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Beginning in the late 1920s, it was realized that there were better clocks than the variable earth spin period. Accordingly, the highly stable orbital motion of the earth about the sun was adopted as a reference. Thus in 1958, the *ephemeris second* was defined as a fixed fraction of the "tropical year 1900", namely $... | {
"Header 1": "**4.5 Measures of time**",
"Header 2": "Ephemeris second",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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In 1972, *Universal Time Coordinated* (UTC) was adopted to bring together the UT and TAI systems. It is based on the atomic second but it is occasionally adjusted by the addition of an extra second, a *leap second*, to maintain it within 0.9 s of UT. In principle, the leap second could be subtracted, if necessary to ma... | {
"Header 1": "*Universal coordinated time (UTC) and leap seconds*",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Astronomers and others have a need for a continuously running time standard that is never adjusted for irregularities of the earth's rotation. A natural choice for this would be atomic time TAI. In fact, since 1972 this has been the standard for such purposes, except for a constant offset of 32.184 s required to match ... | {
"Header 1": "*Universal coordinated time (UTC) and leap seconds*",
"Header 3": "*Terrestrial time (TT)*",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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According to the GR model of time, a clock deep in a potential well will run more slowly than one less deep. Also, a clock moving at high speed relative to some stationary clocks runs more slowly than the "at rest" clocks it passes. The latter is the *time dilation* effect also encountered in special relativity. A cloc... | {
"Header 1": "*Barycentric times*",
"token_count": 690,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The comparison of observations over many years is not simple because of the varying numbers of days in a year (due to leap years) and the different numbers of days in the several months. Accordingly, a continuously running counting system for days is used in astronomy; these are known as Julian dates (JD). The Julian d... | {
"Header 1": "*Barycentric times*",
"Header 3": "*Julian date (JD)*",
"token_count": 620,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The equatorial coordinate system used for celestial measurements depends on the orientation of the earth, and this is a continuously changing function of time (Section 3.2). The time chosen during some period (usually decades) for the specification of celestial coordinates in catalogs and communications between astrono... | {
"Header 1": "*Epochs for coordinate systems*",
"token_count": 1287,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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*Problem 4.21.* (a) If the entire mass of the sun were compressed into a sphere the size of the earth, what would be its density relative to that of the earth? (This is typical of a *white dwarf* star.) (b) If your scale indicates 80 kg when you weigh yourself on earth, what would it read if you weighed yourself with i... | {
"Header 1": "*Epochs for coordinate systems*",
"Header 3": "*4.2 Gravity*",
"token_count": 2028,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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*Problem 4.31*. Draw a horizon coordinate system for an observer in the northern hemisphere at latitude $+60^{\circ}$ . Show the tracks of five stars: a star that is always north of the observer, a star that never sets, a star that rises and sets in the north and another which rises and sets in the south, and a star n... | {
"Header 1": "*Epochs for coordinate systems*",
"Header 2": "4.3 Apparent motions of stars",
"token_count": 949,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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*Problem 4.41*. (a) Confirm that the four cyclic periods given in the text lead to similar eclipses at intervals of 18 y 11.3 d (or 10.3 d). Explain the role of each period in bringing this about. (b) Show that the length of the saros is about 1350 yr. Hint: consider that the sun is susceptible to an eclipse for $\sim... | {
"Header 1": "*Epochs for coordinate systems*",
"Header 2": "4.3 Apparent motions of stars",
"Header 3": "4.4 Lunar and planet motions – eclipses",
"token_count": 1070,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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*Problem 4.51.* Explain the following statement: "It is fortunate indeed that solar and sidereal clocks are not synchronized. If they were, we could never observe certain regions of the sky". Could this situation occur in another star–planet system? If it could, would space-borne telescopes orbiting the planet make mor... | {
"Header 1": "*Epochs for coordinate systems*",
"Header 2": "4.3 Apparent motions of stars",
"Header 3": "*4.5 Measures of time*",
"token_count": 911,
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**Telescopes** and **antennas** collect photons, and the detectors at their foci record the information content of the radiation, its **intensity** and **polarization** as a function of time, and also its **frequency distribution** and **direction of arrival**. There are several common **configurations of optical teles... | {
"Header 1": "**What we learn in this chapter**",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The systems that extract information from faint signals about distant celestial bodies are the source of essentially all our astronomical knowledge. Telescopes collect and concentrate the radiation, and the instruments at their foci analyze one or more properties of the radiation. The systems used for the various frequ... | {
"Header 1": "**What we learn in this chapter**",
"Header 3": "**5.1 Introduction**",
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All astronomical telescope and detector systems have the same purpose, namely, the study of incoming photons with the maximum possible sensitivity, and with the optimum frequency, timing, and angular resolution. One can not always attain the best possible performance in all these aspects at the same time.
A stellar o... | {
"Header 1": "**5.2 Information content of radiation**",
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Telescopes and antennas are the light collectors of astronomy. They come in varying shapes and sizes that depend in part on the frequency of radiation they are designed to detect. Most systems concentrate the incoming radiation by means of *focusing*. Optical telescopes gather light with a lens or a reflecting surface ... | {
"Header 1": "**5.2 Information content of radiation**",
"Header 3": "**5.3** Image formation",
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The radiation from a very distant point-like star arrives at earth as a parallel beam of light. If the light impinges normally onto a thin (ideal) convex lens (Fig. 1a), a parallel bundle of rays will focus to an on-axis point image in the *focal plane*, a distance $f_L$ (*focal length*) beyond the lens. If the paral... | {
"Header 1": "Focal length and plate scale",
"token_count": 657,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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The rate of energy deposited on a single grain of film, or on the single pixel of a modern electronic imaging device, determines whether a given incident energy flux can be detected in a given time. A large telescope aperture (diameter d) will increase the energy flow onto the detector because a larger part of the inco... | {
"Header 1": "Aperture and deposited energy",
"token_count": 1036,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Optical light may be collected and focused by means of a transmitting lens that refracts the rays as shown in Figs. 1a and 2a. The disadvantage of a lens in astronomy is that the light must traverse the glass which can lead to imperfect focusing due for example to color dependence of the index of refraction (*chromatic... | {
"Header 1": "*Telescope configurations*",
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Mechanical collimators may be used to restrict the regions of the sky that their detectors can "see". One type is simply a set of stacked tubes, like handfuls of soda straws (Fig. 3a). Since the radiation from a particular point on the sky impinges on the entire detector, the signal from the star must contend with back... | {
"Header 1": "*Telescope configurations*",
"Header 3": "*Tubular and modulation collimators*",
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If one uses a detector that locates the position at which a photon strikes its surface, such as a piece of film, or a CCD detector (Section 6.3), other arrangements become possible. For example, a mask with randomly placed "pinholes" may be placed above the detector. A point source will then project a pattern of pinhol... | {
"Header 1": "*Multiple pinhole collimator*",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Most radio astronomy is ground based, e.g., the new (in 2000) steerable-dish 100 m diameter Byrd Green Bank Telescope in West Virginia, the huge (300-m) fixed antenna dish in Arecibo, Puerto Rico, and the cooperative world-wide collection of diverse telescopes that work in concert to obtain extremely high angular resol... | {
"Header 1": "*Some real telescopes*",
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The concept of an *antenna beam* is intrinsic to all astronomy. The beam is simply the portion of the sky observed by the detector at a given time (Fig. 5). For example, in a non-focusing detection system, mechanical collimators might restrict the *fieldof-view* to a circular region on the sky of 0.7◦ radius. The detec... | {
"Header 1": "*Meaning of a \"beam\"*",
"token_count": 721,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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When a beam has a small angular size, closely spaced sources can be better resolved. Even if only one source is in the region, a narrow beam gives less contamination from background radiation from directions adjacent to the source. On the other hand, a broader beam is more efficient if one is searching a large portion ... | {
"Header 1": "*Point spread function*",
"token_count": 945,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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A formal derivation of diffraction sums the effect of wavelets originating at each imaginary segment of an aperture such as that shown in Fig. 6a. The aperture could be the aperture of the primary mirror of a telescope. The wavelets are in phase at the aperture if they originate in a plane wave. They interfere with one... | {
"Header 1": "*Fraunhofer diffraction*",
"token_count": 1509,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Diffraction is the primary factor that limits the resolution of radio telescope beams. For example, a 16-m dish antenna observing at ν = 100 MHz (λ = *c*/ν = 3m) has a diffraction pattern, or half beam width, of
$$\theta_{\min} \approx 1.22 \frac{\lambda}{d} = 1.22 \frac{3}{16} = 0.23 \text{ rad} = 13^{\circ}$$
(radi... | {
"Header 1": "*Fraunhofer diffraction*",
"Header 3": "*Radio resolution*",
"token_count": 306,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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At optical wavelengths, the diffraction limit, or beam size, is much smaller because the wavelengths are much smaller. For $\lambda = 500$ nm and a small telescope of aperture d = 0.40 m,
$$\theta_{\min} \approx 1.22 \frac{500 \times 10^{-9}}{0.40} = 1.5 \times 10^{-6} \text{ rad} = 0.3'' \text{ (optical; } d = 0.4... | {
"Header 1": "*Fraunhofer diffraction*",
"Header 3": "Optical resolution",
"token_count": 526,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Consider a distant point-like star. In the ideal case of no atmosphere, the electromagnetic signal would be a plane wave upon its arrival at the telescope (Fig. 8a). Its image in the focal plane would be an Airy disk of angular size $\sim \lambda/d$ where d
Figure 5.8. Isophase patche... | {
"Header 1": "*Fraunhofer diffraction*",
"Header 3": "Isophase patches and speckles",
"token_count": 2050,
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Saha, "Speckle imaging: a boon for astronomical observations", *Proc. Conf. Young Astrophysicsts of Today's India*, 2001; (b,c) K. Saha and D. Maitra, *Indian J. Phys.*, **75B**, 391 (2001); (d) Balega *et al*., *Astron. Lett*. **23**, 172 (1997)]
extent is almost 3. In fact, it is the summation of 128 short (∼10 ms)... | {
"Header 1": "*Fraunhofer diffraction*",
"Header 3": "Isophase patches and speckles",
"token_count": 232,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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In the technique known as *speckle interferometry*, a series of short exposures (e.g., 10 ms) is used to capture the speckles. Keeping in mind that each speckle is a diffraction limited image of the source, the quality of this image may be enhanced by superimposing lots of the speckles in one exposure and summing the c... | {
"Header 1": "*Speckle interferometry*",
"token_count": 699,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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In an adaptive-optics system, the non-planar wavefront is corrected in real time prior to its arrival at the image plane. If successful, the telescope would then yield point-like images with the theoretical resolution of the telescope. Active wavefront correction has been accomplished with the use of a deformable mirro... | {
"Header 1": "*Deformable mirrors*",
"token_count": 331,
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Such a system requires knowledge of the instantaneous state of the atmosphere on time scales of ∼1 ms. This is done by sensing the instantaneous shape of the wavefront each millisecond. The faint stars studied by many astronomers do not provide sufficient light for this. It is necessary therefore to use a bright refere... | {
"Header 1": "*Sensing the wavefront shape*",
"token_count": 614,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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An entire detection and correction system is shown in Fig. 11. The real star and the laser star are observed with the same optics. After passing the correction mirrors, the
Figure 5.11. Complete adaptive optics system. A small telescope directs laser light to a point high in the atmosph... | {
"Header 1": "*Complete system*",
"token_count": 606,
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*Problem 5.31.* (a) Confirm that the Lick 3-m telescope has plate scale of∼14/mm, given its focal length of 15.2 m. What distance on the plate would correspond to a great circle angle of 1◦? (b) What is the focal ratio R of this telescope? (c) A detectable image of a distant star (i.e., a point source) can be obtained ... | {
"Header 1": "*5.3 Image formation*",
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The **detectors** at the foci of telescopes may be **position-insensitive** such as the classic **photomultiplier** and the simple **proportional counter**. **Position-sensitive** detectors at the focus of a telescope provide an overall field of view that includes many beams (resolution elements). The **charge-coupled ... | {
"Header 1": "Detectors and statistics",
"Header 3": "**What we learn in this chapter**",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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At the focal plane of a telescope, an image is formed. It can be viewed directly by eye in two ways, on a piece of frosted glass placed in the focal plane or through an eyepiece. In the latter case, the focal point of the eyepiece is placed at the image so as to create a parallel beam of small extent (pupil sized) that... | {
"Header 1": "Detectors and statistics",
"Header 3": "**6.1 Introduction**",
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A classic example of a position-insensitive detector is the *photomultiplier tube*, which was long used to measure precisely the brightness of stars in different wavebands within the optical band, the visual (*V*), blue (*B*), and ultraviolet (*U*). The process of making such measurements is known as *photometry*. In t... | {
"Header 1": "Detectors and statistics",
"Header 3": "*Photomultiplier and photometry*",
"token_count": 616,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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X rays may be detected in large *proportional counters*. The detector consists of a metal box of inert gas such as argon or xenon with a fine wire ( $\sim$ 25 $\mu$ m diameter) running down its center. The wire, or *anode*, is held at high positive voltage, +1500 V, relative to the box so there is an electric field pe... | {
"Header 1": "Detectors and statistics",
"Header 3": "Proportional counter",
"token_count": 2022,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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For incident 6.0-keV x rays in argon, this would appear as a secondary peak at 6.0 − 2.4 = 3.6 keV. The strength of the peak depends on the counter geometry as well as on the *fluorescent yield* of the element in question, that is, the probability that a vacancy in the K shell will be filled via the emission of an x ra... | {
"Header 1": "Detectors and statistics",
"Header 3": "Proportional counter",
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"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
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Proportional counters can be constructed to provide the approximate location of the incident x ray, or more specifically, the position of the photoelectric conversion in the gas. This is indicated by the location on the anode of the deposited cloud of charge. This location can be determined by making the anode wire res... | {
"Header 1": "Detectors and statistics",
"Header 3": "Position-sensitive proportional counters",
"token_count": 269,
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In optical astronomy, the *charge-coupled device* (CCD) has almost entirely taken the place of the photographic plate. This is a solid state device that can record the integrated intensity of light falling on it as a function of position on its surface (as does photographic film). The surface of the CCD is divided into... | {
"Header 1": "Charge-coupled device",
"token_count": 264,
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} |
The structure of a small portion of a CCD is shown in the side view of Fig. 3a. It consists mostly of a silicon substrate of depth ∼260 m (lower two shaded regions) doped with impurities to make it a *p-type semiconductor*. The impurity is phosphorus which results in a fixed negative charge throughout the material; the... | {
"Header 1": "*Structure of a CCD*",
"token_count": 763,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Now consider what happens when a CCD is exposed to light from the top. The photons will mostly pass through the thin electrode and insulator and will be absorbed in the depleted layers (Fig. 3a), in both the n-type and p-type silicon. A photon impinging on the silicon lattice will usually eject an electron from an atom... | {
"Header 1": "*Exposure to light*",
"token_count": 390,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
After an appropriate time (seconds or minutes), the exposure is stopped, and the charge in each pixel is carried electrically out of the detector. This is accomplished by varying the potentials of the three sets of electrodes (Fig. 3b). Consider Step 0 to be the state of the three electrodes during the exposure, wherei... | {
"Header 1": "*Exposure to light*",
"Header 3": "*Readout of the image*",
"token_count": 1104,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The CCD detectors are a powerful device for astronomy. They are sensitive across the infrared, optical, ultraviolet, and even x-ray bands. For optical photons they are $\sim$ 70% efficient which means that 70% of the photons are converted to electrons. For comparison, the effective efficiency of the eye and film are e... | {
"Header 1": "Utility in optical astronomy",
"token_count": 756,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
In the case of x rays, an individual photon absorbed in the silicon creates a photoelectron. As in the proportional counter, the relaxation of the affected atom yields additional photons which produce additional photoelectrons. The total energy of all the photoelectrons is comparable to that of the incident x ray. Thes... | {
"Header 1": "*Utility in x-ray astronomy*",
"token_count": 526,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A very different type of position-sensitive device is the *spark chamber* used in gamma-ray astronomy experiments at energies above ∼20 MeV. Figure 5a is a simplified view of the *Energetic Gamma-Ray Experiment Telescope* (EGRET) which orbited on the CGRO. It consisted of several types of detectors and was sensitive to... | {
"Header 1": "**6.4 Gamma-ray instruments**",
"Header 2": "*EGRET experiment*",
"token_count": 210,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
*Plastic scintillator anticoincidence*
The EGRET detection system includes a *plastic scintillator* which forms a dome surrounding the upper portions of the system. Its purpose is to help discriminate against the energetic cosmic ray particles (mostly protons) that are continuously traversing the detector; Fig. 5a. W... | {
"Header 1": "**6.4 Gamma-ray instruments**",
"Header 2": "*EGRET experiment*",
"Header 3": "*Detector subsystems*",
"token_count": 536,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The gamma rays are detected in a bank of 36 spark-chamber modules stacked vertically on one another (Fig. 5a), They are interspersed with foils of tantalum, a material of high atomic number (*Z* = 73). The foils have a high cross section for the conversion of a gamma ray to an *electron–positron pair*, often called sim... | {
"Header 1": "*Spark chamber detection of electron–positron pair*",
"token_count": 876,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The electrons also pass through two plastic "timing" scintillators which are above and below the lower group of spark chambers (Fig. 5a). These detectors operate on the same principle as the anticoincidence scintillator; traversal by the electron pair results in an electrical pulse from the photomultipliers. In this ca... | {
"Header 1": "*Spark chamber detection of electron–positron pair*",
"Header 2": "*Timing scintillation detectors (up–down discrimination)*",
"token_count": 229,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The electrons finally pass into a *crystal scintillator*. This is a high-*Z* material, *sodium iodide* (NaI), that causes the electrons to undergo multiple interactions until they have given up their entire kinetic energies to ionization. The recombinations of the ions and electrons cause it to scintillate, to emit lig... | {
"Header 1": "*Energies and arrival directions*",
"token_count": 241,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The Burst and Transient Source Experiment (BATSE) also flew on CGRO. Its appeal for us here is the power inherent in its simplicity. Its primary objective was to study celestial flashes of gamma rays (*gamma-ray bursts*, GRB) that last only for a few minutes and that had been known but unexplained since 1967. The GRBs ... | {
"Header 1": "*BATSE experiment*",
"token_count": 1208,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
In all scientific studies, the precision with which a quantity is measured is all important. For example, consider the detection or discovery of a weak source. If the uncertainty in the measured intensity is large, one might not be convinced the source was even detected; it could have been a perfectly normal fluctuatio... | {
"Header 1": "**6.5 Statistics of measurements**",
"token_count": 450,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Every instrument has its own characteristic background noise. In the absence of any photons impinging on it, apparent spurious signals will be produced. For example, as noted above, cosmic ray particles will pass through a CCD leaving an image in one or more pixels that could be mistaken for a stellar image. Also, the ... | {
"Header 1": "*Instrumental noise*",
"token_count": 364,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Statistical noise is a term applied to the inherent randomness of certain types of events. Consider the detection of photons from a steady source, one that is not pulsing or flaring. Although there is an average rate of arrival of photons, the actual number detected in a limited time interval will fluctuate from interv... | {
"Header 1": "*Statistical fluctuations – \"noise\"*",
"token_count": 2047,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The Poisson distribution approximates well the normal distribution if the latter has $\sigma = \sqrt{m}$ . Note the slight asymmetry of the Poisson distribution relative to the normal distribution. The standard deviation and full width half maximum widths are shown for the higher normal peak; the two normal curves hap... | {
"Header 1": "*Statistical fluctuations – \"noise\"*",
"token_count": 2048,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
One often bases the "error" on the measured, not the true, number, in this case $\sigma=\sqrt{90}=9.5$ . Here one is adopting the measured value as an approximation of the true mean.
The variance of the normal distribution is obtained through substitution of (3) into the integral form of (7),
$$\sigma^{2} = \int_{... | {
"Header 1": "*Statistical fluctuations – \"noise\"*",
"token_count": 294,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
As noted above, if one sets $\sigma=m^{1/2}$ and expects a large number of events, the Poisson distribution is approximated by the symmetric normal distribution. In this case, one can invoke the normal-distribution probabilities in Table 2, e.g., the probability of exceeding $\pm 3\sigma$ is 0.27%. Thus, if the pix... | {
"Header 1": "Measurement significance",
"token_count": 423,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Statistical arguments can seem convincing even if they are wrong. Here we mention two rather common traps in such arguments. The first is to overlook the effect of repeated measurements, or *multiple trials*. Assume that one makes many measurements and finds a 5σ effect in one of them. There is a probability of a stati... | {
"Header 1": "Measurement significance",
"Header 3": "*Statistical traps*",
"token_count": 702,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The discussion above has pertained to the nature of errors on measured quantities. After the data are taken, one invariably manipulates then to obtain other quantities. For example one may divide the accumulated number of counts by accumulation time to find the rate of photon arrivals. Or, one might subtract the backgr... | {
"Header 1": "Propagation of errors",
"token_count": 806,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The subtraction of background makes use of these tools. Let the expected number of source counts detected in a given time interval $\Delta t$ be S, and let the number of expected background counts in the same or equivalent time interval be B. The on-source measurement will thus yield S + B counts, and an off-source m... | {
"Header 1": "Propagation of errors",
"Header 3": "Background subtraction",
"token_count": 821,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Two limiting cases of (16) may be given, one wherein the background counts are much less than the source counts, and vice versa. We assume again that the accumulation time $\Delta t$ is the same for both on-source and off-source measurements.
The low-background $(B \ll S)$ case gives
$$\frac{S}{\sigma_s} \appro... | {
"Header 1": "Propagation of errors",
"Header 3": "Low and high background limits",
"token_count": 661,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Focusing instruments are essentially low-background systems. The detection of only 3 x-ray photons during an observation, in one resolution element of the focal plane could be highly significant because the background in any given resolution element is so low. If the expected background in that element is only 0.1 coun... | {
"Header 1": "Propagation of errors",
"Header 3": "Bright and faint source observations",
"token_count": 665,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Often one will want to compare the data to some theoretical expectation in order to (*i*) derive some parameter or parameters in the theory or (*ii*) to check whether the data are consistent with the theory (or hypothesis). For example one might measure how many cars come down the street every hour in order to find the... | {
"Header 1": "*Finding parameters and checking hypotheses*",
"token_count": 767,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Comparison of data to theory can be carried out with a procedure known as the least squares fit. Consider the data points and theoretical curves in Fig. 9. Each data point is taken at some value *xi* and has a value and uncertainty *yi* ± σ*<sup>i</sup>* indicated with vertical *error bars*. At each position *xi* , cal... | {
"Header 1": "*Least squares fit*",
"token_count": 913,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
A best-fit theoretical curve is not necessarily consistent with the data. When compared to the expected fluctuations in a set of data, the fit may be too bad (high χ2) or too good (low χ2).
One can compare a theoretical function that might underlie the data to the data points themselves to find if in fact the functio... | {
"Header 1": "*Chi square test*",
"token_count": 2018,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
#### **Problems**
#### 6.2 Position-insensitive detectors
*Problem 6.21.* (Requires material in Section 6.5.) The magnitude of the charge pulse from a proportional counter fluctuates in value from one incident x ray to another, even when the incident x rays all have the same energy, e.g., those obtained from an ... | {
"Header 1": "*Chi square test*",
"token_count": 576,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Problem 6.41. Each of the 36 EGRET spark-chamber modules yields x and y coordinates of the track of an electron that passes through the module (see text). Two particle tracks are the typical consequence of the conversion of a gamma ray (Fig. 5a). In a given module, these lead to two x coordinates and two y coordinates ... | {
"Header 1": "*Chi square test*",
"Header 3": "6.4 Gamma-ray instruments",
"token_count": 200,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Radio astronomers have learned to overcome the limitations of diffraction with **interferometry**, the use of two or more telescopes viewing the same source at the same time. The instantaneous beam of two telescopes is an interference **fringe pattern** on the sky. As the earth rotates, the pattern sweeps across a post... | {
"Header 1": "**What we learn in this chapter**",
"token_count": 444,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Interferometry is an observational technique wherein the interference of electromagnetic waves is used to extract the highest possible angular resolution allowed by basic physical principles. We have seen in Section 5.5 how interference within a single telescope aperture due to atmospheric density fluctuations can lead... | {
"Header 1": "**7.1 Introduction**",
"token_count": 713,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Radio interferometry is based on the interference of electromagnetic waves. Consider the reception of a plane radio wave from a distant point source that is directly above two radio telescopes, separated by a short distance *B* (*baseline*; Fig. 1a). The wavefronts in Fig. 1a arrive at the two telescopes in phase becau... | {
"Header 1": "**7.1 Introduction**",
"Header 3": "Principle of interferometry",
"token_count": 1161,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Let the two equatorial telescopes in Fig. 2a each transmit purely sinusoidal waves with the same frequency in the direction approximately normal to their baseline. The two electromagnetic waves will interfere with one another as do the waves in a *ripple tank* experiment wherein two adjacent probes disturb the water su... | {
"Header 1": "*Transmission of radiation*",
"token_count": 1419,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The response *R* (*t*) in Fig. 2d is a plot of the power (amplitude squared) of the summed electric vectors of the electromagnetic signals received from source A at the two detectors as a function of time as the earth rotates. A low-pass filter averages out the very rapid radio-frequency oscillations shown in the first... | {
"Header 1": "*Transmission of radiation*",
"Header 3": "*Earth rotation*",
"token_count": 281,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
How is the response *R* (*t*) used to provide information about the location of a point source of unknown celestial position? Knowledge of the physical locations of the telescopes in inertial space (derived from their location on the earth and the angle of the earth's rotation) allows one to locate precisely the positi... | {
"Header 1": "*Position of source*",
"token_count": 1464,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Heretofore, we considered only sources nearly overhead, i.e., at $\theta \approx 90^{\circ}$ ; this angle is defined in Fig. 1c. The location of the great and small circles of constructive
Figure 7.5. (a) Geometry which defines the directions of 100% visibility for two telescopes separ... | {
"Header 1": "All-sky fringe pattern",
"token_count": 1147,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The two telescopes sample the incoming electromagnetic wave at two places in the plane wavefront. The two telescopes are configured to select the same component of the incoming (vector) *E* field (the same *linear polarization*); thus the measured scalar components *E*<sup>1</sup> and *E*<sup>2</sup> can be used to des... | {
"Header 1": "*Wavefront samples*",
"token_count": 517,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Electric fields are additive; two fields at a given position and time yield a net field obtained by vector addition. After detection of the two wave samples from our two telescopes, the fields could be added by bringing them together with coaxial cables. In our case (same polarizations), one may add the scalar componen... | {
"Header 1": "*Wavefront samples*",
"Header 3": "*Summed waves*",
"token_count": 1270,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The same variation can be obtained if the waves are multiplied rather than added. To motivate this, take the square of the summed waves (9) directly,
$$E^{2} = [E_{0}\cos\omega t + E_{0}\cos(\omega t - \phi)]^{2}$$
(7.15)
$$E^{2} = E_{0}^{2} [\cos^{2} \omega t + 2\cos \omega t \cos(\omega t - \phi) + \cos^{2}(\omeg... | {
"Header 1": "*Wavefront samples*",
"Header 3": "Multiplied waves",
"token_count": 1478,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The basic elements of the signal processing are illustrated in Fig. 5a. It is rather difficult to work directly with radio frequencies, so, as in ordinary radio receivers, the frequency from each telescope is converted to a lower *intermediate frequency* (IF) provided by a local oscillator which beats against the ampli... | {
"Header 1": "Signal processing",
"token_count": 494,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
At a given time, the angular spacing θ (rad or rad/cycle) between adjacent visibility lines (on the sky) of a single antenna pair is known as the *spatial wavelength*. The inverse wavelength (θ) <sup>−</sup><sup>1</sup> is the *spatial frequency* (cycles/radian or rad−1), that is, the number of visibility lines per rad... | {
"Header 1": "*Spatial frequencies*",
"token_count": 928,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
One can visualize the Fourier (or u, v) plane as the plane tangent to (and fixed to) the celestial sphere near the center of the source region (Fig. 7b). As noted, this tangent point is called the phase center. The phase center is in the direction s (Fig. 7a) and is fixed on the celestial sphere (unlike the zero phase ... | {
"Header 1": "*Spatial frequencies*",
"Header 3": "Projected baseline",
"token_count": 670,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The shading method is a simple extension of the line-drawing method of Fig. 4. Instead of drawing lines at times of maximum response as in Fig. 4, the entire sinusoidal two-telescope point-source fringe pattern (i.e.,the two-telescope beam) is binned (shaded) onto a map of the sky as shown in Fig. 8a. This is done for ... | {
"Header 1": "*Cross-correlation or \"shading\" method*",
"Header 3": "*Bins on the sky*",
"token_count": 833,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Formally, in the shading method, the quantity that is summed for a given sky position α,δ at time *t* is the product of: (*i*) the fringe pattern on the sky (e.g., Fig. 8a), which is the interferometric response at time *t* to a unit point source at α,δ, or from (14), *R*PS (α,δ,*t*) = 1 + cos φ(α,δ,*t*), and (*ii*) th... | {
"Header 1": "*Cross-correlation*",
"token_count": 1169,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Finally, let us note a problem with this shading method as described here; it weights all time intervals equally. This can lead to uneven shading if some Fourier
components are sampled more than others. For example, consider the 12-hour northpole observation of Figs. 3 and 4c. Suppose that during half of this time, t... | {
"Header 1": "*Cross-correlation*",
"Header 3": "Equal weighting of time intervals",
"token_count": 306,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
Imagine that the brightness of the sky actually varies as a one-dimensional sinusoid that varies along the equator as a cosine wave, e.g., (1 + cos φ)/2, in the E–W direction, and with no variation in the N–S direction. It would resemble ocean waves frozen in time, like the waves of Fig. 8a. Let the spatial period of t... | {
"Header 1": "*Principle of aperture synthesis*",
"token_count": 926,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
The response functions illustrated heretofore were for point sources only. Here we present the response to an extended source of arbitrary brightness distribution. Let the unit source vector *s* refer to the direction of the phase center (tangent point of the Fourier plane, Fig. 7b). The expression (21) gives the respo... | {
"Header 1": "*Arbitrary sky brightness distribution*",
"token_count": 2027,
"source_pdf": "datasets/websources/Astronomy_v1/Astronomy/Hale Bradt_2004.pdf"
} |
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