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chapter 1

Course: ASTR 110G, Fall 2009
School: NMSU
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1 Chapter Our Place in the Universe 1.1 Our Modern View of the Universe Our goals for learning: What is our place in the universe? How did we come to be? How can we know what the universe was like in the past? Can we see the entire universe? What is our place in the universe? Our Cosmic Address Star A large, glowing ball of gas that generates heat and light through nuclear fusion Planet Mars Neptune A...

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1 Chapter Our Place in the Universe 1.1 Our Modern View of the Universe Our goals for learning: What is our place in the universe? How did we come to be? How can we know what the universe was like in the past? Can we see the entire universe? What is our place in the universe? Our Cosmic Address Star A large, glowing ball of gas that generates heat and light through nuclear fusion Planet Mars Neptune A moderately large object that orbits a star; it shines by reflected light. Planets may be rocky, icy, or gaseous in composition. Moon (or satellite) An object that orbits a planet. Ganymede (orbits Jupiter) Asteroid A relatively small and rocky object that orbits a star. Ida Comet A relatively small and icy object that orbits a star. Solar (Star) System A star and all the material that orbits it, including its planets and moons Nebula An interstellar cloud of gas and/or dust Galaxy A great island of stars in space, all held together by gravity and orbiting a common center M31, The Great Galaxy in Andromeda Universe The sum total of all matter and energy; that is, everything within and between all galaxies How did we come to be? How can we know what the universe was like in the past? Light travels at a finite speed (300,000 km/s). Destination Moon Sun Sirius Andromeda Galaxy Light travel time 1 second 8 minutes 8 years 2.5 million years Thus, we see objects as they were in the past: The farther away we look in distance, the further back we look in time. Example: This photo shows the Andromeda Galaxy as it looked about 2 1/2 million years ago. Question: When will we be able to see what it looks like now? M31, The Great Galaxy in Andromeda Definition: Light-Year The distance light can travel in one year. About 10 trillion kilometers (6 trillion miles). At great distances, we see objects as they were when the universe was much younger. Can we see the entire universe? Thought Question Why cant we see a galaxy 15 billion light-years away? (Assume the universe is 14 billion years old.) Because no galaxies exist at such a great distance. Galaxies may exist at that distance, but their light would be too faint for our telescopes to see. Because looking 15 billion light-years away means looking to a time before the universe existed. Thought Question Why cant we see a galaxy 15 billion light-years away? (Assume the universe is 14 billion years old.) Because no galaxies exist at such a great distance. Galaxies may exist at that distance, but their light would be too faint for our telescopes to see. Because looking 15 billion light-years away means looking to a time before the universe existed. What have we learned? What is our place in the universe? Earth is part of the Solar System, which is the Milky Way Galaxy, which is a member of the Local Group of galaxies in the Local Supercluster. How did we come to be? The matter in our bodies came from the Big Bang, which produced hydrogen and helium. All other elements were constructed from H and He in stars and then recycled into new star systems, including our solar system. What have we learned? How can we know what the universe was like in the past? When we look to great distances we are seeing events that happened long ago because light travels at a finite speed. Can we see the entire universe? No, the observable portion of the universe is about 14 billion light-years in radius because the universe is about 14 billion years old. 1.2 The Scale of the Universe Our goals for learning: How big is Earth compared to our solar system? How far away are the stars? How big is the Milky Way Galaxy? How big is the universe? How do our lifetimes compare to the age of the universe? How big is Earth compared to our solar system? Lets reduce the size of the solar system by a factor of 10 billion; the Sun is now the size of a large grapefruit (14 cm diameter). How big is Earth on this scale? A. B. C. D. an atom a tip of a ball point pen a marble a golf ball Lets reduce the size of the solar system by a factor of 10 billion; the Sun is now the size of a large grapefruit (14 cm diameter). How big is Earth on this scale? A. B. C. D. an atom a tip of a ball point pen a marble a golf ball The scale of the solar system On a 1-to-10 billion scale: Sun is the size of a large grapefruit (14 cm) Earth is the size of a tip of a ball point pen, 15 meters away. Relative Distance of the Nearest Star How far away are the stars? On our 1-to-10 billion scale, its just a few minutes walk to Pluto. How far would you have walk to to reach Alpha Centauri? A. B. C. D. 1 mile 10 miles 100 miles the distance across the United States (2500 miles) Answer: D, the distance across the United States Relative Distance of the Nearest Star How big is the Milky Way Galaxy? The Milky Way has about 100 billion stars. On the same ten billion-toone scale The Size of the Milky Way Thought Question Suppose you tried to count the more than 100 billion stars in our galaxy, at a rate of one per second How long would it take you? A. A few weeks B. A few months C. A few years D. A few thousand years Thought Question Suppose you tried to count the more than 100 billion stars in our galaxy, at a rate of one per second How long would it take you? A. A few weeks B. A few months C. A few years D. A few thousand years How big is the Universe? The Milky Way is one of about 100 billion galaxies. 1011 stars/galaxy 1011 galaxies = 1022 stars It has as many stars as grains of (dry) sand on all Earths beaches. Now lets step through the Universe in powers of 10: Zooming Out or Zooming In 26 Orders of Magnitude How do our lifetimes compare to the age of the Universe? The Cosmic Calendar: A scale on which we compress the history of the universe into 1 year. What have we learned? How big is the Earth compared to our solar system? On a scale of 1-to-10 billion, the Sun is about the size of a grapefruit. The Earth is the size of a tip of a ball point pen about 15 m away. The distances between planets are huge compared to their sizes. How far away are the stars? On the same scale, the stars are thousands of kilometers away. How big is the Milky Way Galaxy? It would take more than 3,000 years to count the stars in the Milky Way Galaxy at a rate of one per second. The Milky Way Galaxy is about 100,000 light-years across. What have we learned? How big is the universe? 100 billion galaxies in the observable universe 14 billion light-years in radius As many stars as grains of sand on Earths beaches How do our lifetimes compare to the age of the universe? On a cosmic calendar that compresses the history of the universe into one year, human civilization is just a few seconds old, and a human lifetime is a fraction of a second. 1.3 Spaceship Earth Our goals for learning: How is Earth moving in our solar system? How is our solar system moving in the Milky Way Galaxy? How do galaxies move within the universe? Are we ever sitting still? How is Earth moving in our solar system? Contrary to our perception, we are not sitting still. We are moving with the Earth in several ways, and at surprisingly fast speeds. Earth rotates around its axis once every day. Earth orbits the Sun (revolves) once every year at an average distance of 1 AU 150 million km. with Earths axis tilted by 23.5 (pointing to Polaris). and rotates in the same direction it orbits, counterclockwise as viewed from above the North Pole. Our Sun moves randomly relative to the other stars in the local Solar neighborhood at typical relative speeds of more than 70,000 km/hr. but stars are so far away that we cannot easily notice their motion. And it orbits the galaxy every 230 million years. More detailed study of the Milky Ways rotation reveals one of the greatest mysteries in astronomy Most of Milky Ways light comes from disk and bulge . but most of the mass is in its halo How do galaxies move within the universe? Galaxies are carried along with the expansion of the universe. But how did Hubble figure out that the universe is expanding? Hubble discovered that all galaxies outside our Local Group are moving away from us. the more distant the galaxy, the faster it is racing away. Conclusion: We live in an expanding universe. Are we ever sitting still? Earth rotates on axis: > 1,000 km/hr Earth orbits Sun: > 100,000 km/hr Solar system moves among stars: ~ 70,000 km/hr Milky Way rotates: ~ 800,000 km/hr Milky Way moves in Local Group Universe expands What have we learned? How is Earth moving in our solar system? It rotates on its axis once a day and orbits the Sun at a distance of 1 AU = 150 million km How is our solar system moving in the Milky Way Galaxy? Stars in the Local Neighborhood move randomly relative to one another and orbit the center of the Milky Way in about 230 million years What have we learned? How do galaxies move within the universe? All galaxies beyond the Local Group are moving away from us with expansion of the universe: the more distant they are, the faster theyre moving Are we ever sitting still? No!
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper B Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper F Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper H Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
Cambridge College - MATH - c3
FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper I Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
Cambridge College - MATH - c3
FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper J Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
Cambridge College - MATH - c3
FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper K Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
Cambridge College - MATH - c3
FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper L Time: 1 hour 30 minutesInstructions and InformationCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration.
Cambridge College - MATH - c3
FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper BMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper CMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper DMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper EMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper FMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper GMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper HMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper IMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper JMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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FOR EDEXCELGCE Examinations Advanced SubsidiaryCore Mathematics C4Paper KMARKING GUIDEThis guide is intended to be as helpful as possible to teachers by providing concise solutions and indicating how marks could be awarded. There are obviously altern
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Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper A Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper B Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper C Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper D Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper E Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper F Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper G Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper H Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper I Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper J Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper K Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Core Mathematics C4 Advanced LevelPaper L Time: 1 hour 30 minutesInstructions and InformationFor EdexcelCandidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Full marks may be obtain
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper A1 2 3x = + (a) (x - 1)(x + 2) x-1 x+233.(a)dy = ex - 3 dx at M ex = 3 x = ln 3ln 3 ln3 3 (ex - 3x)dx = ex - x 2 2 01.(using `cover up' rule)(3)(2)(b)22 1 + dx = ln(x - 1) + 2 ln(x + 2) x-1 x+2 5 4 25 = ln 2
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper B1. (a) xy - 2y + 5x - 10 = 12 x dy dy + y.1 - 2 +5=0 dx dx (b) - 3. 1 1 - 1 2 2 (a) 1 + (8x) + 2 2 1 1 3 - - 2 2 2 (8x)2 + 3.2 (8x)3 + . . . (3) (1)= 1 + 4x - 8x 2 + 32x 3 1 1 <x< 8 8dy (x - 2) = -(y + 5) dx dy y+5 =
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper C1. (a) 5x + 7 2 3 = + (x + 1)(x + 2) x+1 x+2 y = 2(x + 1)-1 + 3(x + 2)-1 dy = -2(x + 1)-2 - 3(x + 2)-2 dx d2 y dx 2 = 4(x + 1)-3 + 6(x + 2)-3 d2 y dx 2 4 6 13 = 3 + 3 = 18 2 3 (3) 4. (3) (b) valid for - 1 1 <x< 2 2 (1)
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper D1. (a)y3.GivendS dr = 640 cm2 s-1 . To find . dt dtS = 4r 2 dS = 8r dr2 xdS dr dS = dt dr dt when r = 5, (2) 640 = 8 5 640 dr = dt 40 = 16 cm s-1 (4) dr dt22 2(b) volume = y 2 dx = (9 - x 2 )dx0 02 1 8 = 9x
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper E3. 1. (a) when y = 1, 4x 2 + 3 = 12 x2 = 9 4 3 2 dy 8x 4x =- =- dx 6y 3y (2) 1 2 cos 2 cos 2 dy 2 = =- (a) dx - sin sin 1 cos 3 = - 2 = -1 at = , gradient = - 1 6 sin 6 2 3 1 1 (b) at = , x = cos = and y = sin = 3. 6 6
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper F1. dx dy = cos t, = 2 + sin t dt dt cos t dy = dr 2 + sin t dy = 0. stationary points where dx 3 , . 2 2 when t = , x = 2 - cos = ; y = 2 2 2 2 i.e. cos t = 0 t = t= 3 , 2 x =2 3 3 - cos = 3; 2 2 y = 1 + sin 3 =0 2 (5)
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper G1. (a) 2 cos t dy = dx - sin t when t = , 2 gradient = 0 (2) (b) (1 + bx) 1 + 6ax + 15a 2 x 2 = 1 + 6ax + 15a 2 x 2 + bx + 6abx 2 we have 6a + b = -9 15a 2 + 6ab = 24 from equation [A] substitute in [B] Hence a = -2, .[
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper Hx 1. (a) 1 1 + e-1 -1 1 1+e 0 1 1+1 1 1 1+ 1 e e = 1 e+1 1+ e 1 2. (a) (i) differentiating implicitly, 1 = ey 1 1 dy = y = dx e x (ii) when y = 0, x = e0 = 1 dy =1 dx dy dx (2)integral 1 e 1 1 + +2 2 1+e e+1 2 (b) (4)
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper I1. (a) dy 1 dx = (- sin ), = 2 cos 2 d (1 + cos ) d dy - sin = dx (1 + cos )2 cos 2 where = , gradient = 6 1 - 1 2 =- 1 3 3 2 2 1+ 1+ 2 2 2 1 (2 - 3) =- =- = 3-2 2+ 3 (2 + 3)(2 - 3) 3. (a) 1 dy + 3x 2 - 2 = 0 y dx dy =
Cambridge College - MATH - C4
Worked Solutions Edexcel C4 Paper J1. (a) We are given that dA = 40 cm2 s-1 dt 3. (a) 3 2 + 2x - 1 x + 2 1 dy = y (using `cover up' rule) (3)(b)after 10 seconds area of circle = 400 cm2 so r 2 = 400 r= (b) A = r 2 dA = 2r dr dA dr dA = dt dr dt when r