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Download AN INTRODUCTION TO ASTRONOMY Dr. Uri Griv Department of Physics, Ben-Gurion University
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BEN-GURION UNIVERSITY AN INTRODUCTION TO ASTRONOMY Dr. Uri Griv Department of Physics, Ben-Gurion University Tel.: 08-6428226 Email: [email protected] Astronomy Picture of the Day Discover the cosmos! Each day a different image or photograph of our fascinating universe is featured, along with a brief explanation written by a professional astronomer. 2001 October 4 M74: The Perfect Spiral Credit: Gemini Observatory, GMOS Team Explanation: If not perfect, then this spiral galaxy is at least one of the most photogenic. An island universe of about 100 billion stars, 30 million light-years away toward the constellation Pisces, NGC 628 or M74 presents a gorgeous face-on view to earthbound astronomers. Classified as an Sc galaxy, the grand design of M74’s graceful spiral arms traced by bright blue star clusters and dark cosmic dust lanes, is similar in many respects to our own home galaxy, the Milky Way. Recorded with a 28 million pixel detector array, this impressive image celebrates first light for the Gemini Multi-Object Spectrograph (GMOS), a state-of-the-art instrument now operational at the 8-meter Gemini North telescope. The Gemini North Observatory gazes into the skies above Mauna Kea, Hawaii, USA, while its twin observatory, Gemini South, is scheduled to begin operations later this year from Cerro Pachón in central Chile. Tomorrow’s picture: A Flock of Stars The Birth of Astronomy • To most people astronomy means stars • The Solar system: (a) all the planetary orbits are nearly circular and (b) lie within only several degrees of the plane of ecliptic 2 The Birth of Astronomy • The radius of the Earth: Erastothenes’ method • SUGGESTION: The Sun is far enough away that the rays from the Sun are virtually parallel when they strike the Earth 3 The Birth of Astronomy • Aristarchos’ method: the relative distance of the Moon and the Sun • SUGGESTION: The angular sizes of the Moon and the Sun do not change appreciably with time → they maintain nearly constant distances from the Earth (in other words, the orbits are circular) 4 The Birth of Astronomy • The reason for the seasons 5 The Birth of Astronomy • The signs of the Zodiac 6 The Birth of Astronomy • The tides 7 The Tides 8 The Birth of Astronomy – Tides GMe Mm GMe Mm + FS − FC = − 2 (R − re ) R2 re ≈ −2GMe Mm 3 R 9 Tidal Instability 10 Accretion Disks • Roche lobe 11 Kepler’s Three Laws • a31 P12 = 3 2 P2 a2 12 Kepler’s Laws – 1st Law 13 Kepler’s Laws – 2nd Law 14 Kepler’s Three Laws – 2nd Law • Consider a planet Mp moving about the sun in an elliptical orbit • The gravitational force F ||r • Then the torque acting on the planet due to this central force τ = r×F = 0 • Recall that the torque equals the time rate of change of angular momentum L, or τ = dL/dt. Therefore, because τ = 0 L = r×p = mr×v = const (1) • Since L = const, the planet’s motion at any instant is restricted to the plane formed by r and v 15 Kepler’s 2nd Law • The radius vector r sweeps out an area dA in a time dt • Since dr = vdt, we get 1 1 L dA = |r×dr| = |r×vdt| = dt 2 2 2Mp so L dA = = const → Kepler′ s second law dt 2Mp • The law applies to any central force, whether inverse-square or not • PROOV: Mp va ra = Mp vp rp , where the subscripts a and p represent aphelion and perihelion, respectively 16 Kepler’s Three Laws – 3rd Law • Consider a planet Mp in a circular orbit about the sun of mass Msun • The gravitational force on the planet is equal to the centripetal force GMpMsun Mp v 2 = 2 r r (2) • But the orbital velocity v = 2πr/T , then (2πr/T )2 GMsun = r2 r • Finally, 2 T = where Ksun = = const 2 4π GMsun 4π 2 GMsun r3 = Ksunr3 , (3) = 2.97 × 10−19 s2 /m3 17 Kepler’s Three Laws – 3rd Law • Kepler’s 3rd law is obtained from the inverse-square law for circular orbits, F ∝ 1/r2 • Useful planetary data • Note that the Pluto is no longer a planet 18 Kepler’s Three Laws – 3rd Law • EXAMPLE: Calculate the mass of the sun using Tearth = 3.156 × 107 s and Rearth = 1 AU = 1.496 × 1013 cm. • Using Eq. (3), we get Msun 3 4π 2 Rearth 33 = = 1.99 × 10 g 2 GTearth • Note that the sun is 333 000 times as massive as the earth ! 19 Possible Planetary Orbits 20 Planetary Orbits • Possible Earth’s orbits, Vrot = 0 − 50 km/s 21 Eccentricity • Two elliptical and one circular orbits; the same major axis 22 Gravitational Potential Energy • The energy associated with the position of a particle m • The change in the gravitational potential energy Z rf Z r2 dr F (r)dr = GMe m Uf − Ui = − 2 ri r r1 • Uf − Ui = −GMe m r1i − r1f • Taking Ui = 0 at ri = ∞, we obtain GMe m U (r) = − negative! r 23 Gravitational Potential Energy • The gravitational potential energy associated with any pair of separated particles m1 and m2 Gm1 m2 U =− r • Binding energy Gm1 m2 /r • That is, U becomes less negative as r increases • Consider Earth–particle system: Total energy is const 1 2 GMe m E = mv − 2 r • Newton’s 2nd law: GMe m GMe m mv 2 1 2 = → mv = r2 r 2 2r 24 (4) Conservation of Energy • Substituting this into Eq. (4), GMe m GMe m em E = GM − = − ≡ 2r r 2r U 2 • E is negative ! • EXAMPLE: Calculate the work required to move an earth satellite m from a circular orbit of radius 2Re to one of radius 3Re • GMe m Initial energy Ei = − 4Re em Final energy Ef = − GM 6Re em Work W = Ef − Ei = GM 12Re 25 Conservation of Energy • EXAMPLE: Escape velocity • An object m is projected vertically upward from the earth’s surface with vi 26 Escape Velocity • When the object reaches its maximum altitude, vf and rf = rmax • Because the total energy of the system is conserved, 1 2 GMe m GMe m mvi − =− 2 Re rmax 1 • → vi2 = 2GMe R1e − rmax • From this expression we can find h = rmax − Re • Settingq rmax = ∞ and vi = vesc , we get e vesc = 2GM Re – independent of m 27 Escape Velocity • Escape velocities from planets • Atmosphera 28 First Cosmic Velocity • First cosmic velocity v = • If r ≈ re , v ≈ 8 km/s 29 q GMe r Libration Points 30 Conservation of Energy 1 1 2 total energy = E = m1 v1 + m2 v22 2 2 Gm1 m2 + − = const r • The bound elliptical orbit • First cosmic velocity: v = p GM/r p • Second cosmic velocity: v = 2GM/r 31 Conservation of Energy • The orbits • The system is bound: E < 0 • Unbounded orbits: E > 0 • The transition case: E = 0 32 Angular Momentum – Conservation • For a single particle, j = r×p, where p = mv 33 Angular Momentum – Conservation • EXAMPLE: j = m(r×v) = const 34 Angular Momentum – Conservation • EXAMPLE: Solar system j = m(r×v) = const 35