Astronomy Principles and Practice Fourth Edition.pdf

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Stationary points 159 Figure 12.8. The velocities of the Earth and a superior planet produce retrograde motion of the planet (P 1 )at opposition and direct motion of the planet (P 2 ) at quadrature. At the following quadrature, the positions of planet and Earth are P 2 and E 2 ,whereSE 2 P 2 = 90 ◦ . The Earth’s orbital velocity V ⊕ is now along the line P 2 E 2 but the planet’s velocity V P has a component V P sin α, perpendicular to E 2 P 2 . The other component V P cos α lies along the line P 2 E 2 and, like the Earth’s velocity V ⊕, does not contribute to the observed angular velocity of the planet. This geocentric angular velocity at quadrature is, therefore, V P sin α E 2 P 2 and is seen to be in the same direction as the orbital movement. It is thus direct at quadrature. 12.9 Stationary points Some time between opposition and quadrature, the planet’s geocentric angular velocity must change from being retrograde to being direct. The point where it is neither retrograde nor direct is said to be a stationary point. We wish to obtain an expression for the elongation E at a stationary point in terms of the distances of planet and Earth from the Sun. From such an expression and the measured value of the elongation, the planet’s heliocentric distance can be calculated. It is also possible to obtain an expression giving the value of the angle θ between the heliocentric radius vectors of the planet and Earth when the stationary point is reached, in terms of the two planets’ distances b and a respectively. This expression, with a knowledge of the synodic period of the planet, enables a prediction to be made of the time of the next stationary point after an opposition of the planet. In figure 12.9, let the positions of Earth and planet at a stationary point be E and P. The velocity of P relative to E must lie along the geocentric radius vector EP if the planet appears stationary. This velocity is represented by the line PB; the parallelogram PABC is the parallelogram of velocities where PA = V P , PC =−V ⊕ and PB is the resultant of these velocities. Since the orbits are circular, the angle θ between the heliocentric radius vectors must be the angle PCB between the velocity vectors. Also it is readily seen that APB = 90 + φ. Extend CP to meet the line SE produced at D. ThenDPA = θ and EPD = 90 − (θ + φ). Hence, in △PCB, using the sine formula, we have sin[90 − (θ + φ)] V P = sin(90 + φ) V ⊕
160 Geocentric planetary phenomena Figure 12.9. Superior planet at a stationary point. or cos(θ + φ) = V P cos φ. (12.6) V ⊕ Now in any plane triangle XYZ with sides x, y, z and opposite angles X, Y , Z,wehave x = y cos Z + z cos Y. Applying this well-known formula to △SEP, we may write SP = PEcos φ + SE cos θ i.e. b = PEcos φ + a cos θ. (12.7) Applying it again to △SEP, we obtain SE = SP cos θ + PEcos SEP or a = b cos θ − PEcos(θ + φ) (12.8) since SEP = 180 − (θ + φ). Equations (12.7) and (12.8) give b − a cos θ = PEcos φ b cos θ − a = PEcos(θ + φ) so that b − a cos θ b cos θ − a = cos φ cos(θ + φ) . (12.9)
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Chapter 1 Naked eye observations 1.
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A month 5 seen to rise above the ea
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A year 7 between south and west. An
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Chapter 2 Ancient world models Firs
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Ancient world models 11 Figure 2.1.
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Chapter 3 Observations made by inst
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Instrumentation in astronomy 15 Fig
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The role of the observer 17 Earth a
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Electromagnetic radiation 19 meteor
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Electromagnetic radiation 21 If, fo
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Electromagnetic radiation 23 device
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Direction of arrival of the radiati
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Brightness 27 Figure 5.3. The windo
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Brightness 29 physical conditions w
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Time 31 to house radio transmitters
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Chapter 6 The night sky 6.1 Star ma
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Simple observations 35 Figure 6.1.
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Chapter 7 The geometry of the spher
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Position on the Earth’s surface 4
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Position on the Earth’s surface 4
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Spherical trigonometry 51 determine
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Spherical trigonometry 53 Figure 7.
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The small spherical triangle 55 7.4
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Problems—Chapter 7 57 Although th
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Chapter 8 The celestial sphere: coo
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The equatorial system 61 Figure 8.2
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Circumpolar stars 63 Figure 8.4. A
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Also Hence, we have or The measurem
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The geocentric celestial sphere 67
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Transformation of one coordinate sy
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Right ascension 71 or cos 47 ◦ 39
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The Sun’s geocentric behaviour 73
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Sunset and sunrise 75 Figure 8.15.
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Megalithic man and the Sun 77 Figur
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The ecliptic system of coordinates
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Galactic coordinates 81 Table 8.1.
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Galactic coordinates 83 Figure 8.22
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Galactic coordinates 85 Figure 8.25
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Problems—Chapter 8 87 where ε is
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Sidereal time 89 Figure 9.1. The ti
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Sidereal time 91 Figure 9.3. An equ
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Mean solar time 93 Figure 9.4. Posi
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The relationship between mean solar
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The civil day and timekeeping 97 It
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The tropical year and the calendar
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The Earth’s geographical zones 10
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Hence, the period of the year durin
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Twilight 105 Figure 9.10. The heati
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Twilight 107 For this to happen, ZB
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Page 120 and 121: Geocentric parallax 121 Figure 10.6
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Page 130 and 131: Problems—Chapter 10 131 Recent ob
Page 132 and 133: Chapter 11 The reduction of positio
Page 134 and 135: The velocity of light 135 Table 11.
Page 136 and 137: The constant of aberration 137 Figu
Page 138 and 139: Diurnal and planetary aberration 13
Page 140 and 141: Precession of the equinoxes 141 Fig
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Page 144 and 145: Nutation 145 Figure 11.10. The grav
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Page 148 and 149: Chapter 12 Geocentric planetary phe
Page 150 and 151: The Copernican System 151 Figure 12
Page 152 and 153: Planetary configurations 153 (a) V
Page 154 and 155: The synodic period 155 Figure 12.5.
Page 156 and 157: Measurement of planetary distances
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Page 164 and 165: Problems—Chapter 12 165 with an o
Page 166 and 167: Planetary orbits 167 Figure 13.1. M
Page 168 and 169: Planetary orbits 169 If P 1 SP 2 is
Page 170 and 171: Newton’s law of gravitation 171 a
Page 172 and 173: The Principia of Isaac Newton 173 N
Page 174 and 175: The two-body problem 175 Figure 13.
Page 178 and 179: The two-body problem 179 13.6.5 The
Page 182 and 183: The astronomical unit 183 For an ex
Page 184 and 185: Chapter 14 Celestial mechanics: the
Page 186 and 187: 14.3 General properties of the many
Page 188 and 189: Special perturbation theories 189 F
Page 190 and 191: Special perturbation theories 191 F
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Page 194 and 195: The geostationary satellite 195 in
Page 196 and 197: Interplanetary transfer orbits 197
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Page 204 and 205: Problems—Chapter 14 205 (figure 1
Page 206 and 207: 210 The radiation laws Figure 15.1.
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212 The radiation laws Table 15.1.
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214 The radiation laws Figure 15.2.
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216 The radiation laws In order to
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218 The radiation laws where σ is
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220 The radiation laws The brightne
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222 The radiation laws centrifugal
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226 The radiation laws is moving pa
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230 The radiation laws In most phys
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232 The radiation laws radiation ha
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234 The radiation laws radiation is
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236 The radiation laws Figure 15.14
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Chapter 16 The optics of telescope
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240 The optics of telescope collect
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Chapter 17 Visual use of telescopes
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274 Visual use of telescopes By sub
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276 Visual use of telescopes 17.3 M
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278 Visual use of telescopes By let
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280 Visual use of telescopes Figure
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282 Visual use of telescopes For sm
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284 Detectors for optical telescope
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306 Astronomical optical measuremen
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Chapter 20 Modern telescopes and ot
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332 Modern telescopes and other opt
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Chapter 21 Radio telescopes 21.1 In
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354 Radio telescopes Figure 21.2. A
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358 Radio telescopes Figure 21.6. T
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362 Radio telescopes (a) Paraboloid
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364 Radio telescopes Figure 21.12.
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366 Radio telescopes The record fro
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368 Radio telescopes 2 N N Figure 2
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Chapter 22 Telescope mountings 22.1
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376 Telescope mountings arc and thi
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378 Telescope mountings Figure 22.4
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380 Telescope mountings The design
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382 High energy instruments and oth
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404 Practical projects to give the
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406 Practical projects Figure 24.2.
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408 Practical projects measurement
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416 Practical projects Summer Time
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418 Practical projects Figure 24.12
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422 Practical projects point was ch
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426 Practical projects N φ Earth 1
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428 Practical projects 24.5 Solar d
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430 Practical projects allows the S
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432 Practical projects 24.5.4 The e
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438 Practical projects Now in t hou
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440 Practical projects Table 24.2.
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442 Practical projects 24.7.2 The i
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448 Practical projects right ascens
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450 Practical projects amenable to
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452 Web sites • W 20.5—www.mrao
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Appendix: Astronomical and related
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456 Appendix: Astronomical and rela
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Bibliography 1 Source books The Ast
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Answers to problems Chapter 7 1. 32
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462 Answers to problems Figure A8.2
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464 Answers to problems Figure A10.
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466 Answers to problems Figure A13.
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468 Answers to problems 5. 1·64 6.
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470 Index black body radiation, 216
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472 Index atom, 221 line, 353 illum
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474 Index potential energy, 177 pow
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