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Extragalactic Astronomy & Cosmology First-Half Review [4246] Physics 316

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Jane Turner [4246] PHY 316 (2003 Spring) Kepler vs Newton

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Jane Turner [4246] PHY 316 (2003 Spring) Newton’s Laws An inertial frame is one in which under the influence of no forces, an object will remain at rest or in uniform motion Newton formulated a theory of mechanics & gravity that explained the solar system with remarkable accuracy! Realized that gravity responsible for the motion of the Moon and planets. Newton’s law of universal gravitation Every mass attracts every other mass Force drops off with square of distance Kepler’s laws are a direct consequence of Newton’s law of gravity

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Jane Turner [4246] PHY 316 (2003 Spring) Lecture 3 Law of Universal Gravitation Law of universal gravitation F = GMm/d 2 Remember this one! Gravitational force follows an inverse square law- doubling separation between two objects, grav attraction drops x 4

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Jane Turner [4246] PHY 316 (2003 Spring) Lecture 3 Newtons form of Keplers Third Law Newton also generalized Kepler’s third law as P 2 =4 2 R 3 /G(M 1 +M 2 ) Allowed Kepler’s Laws to be applied to moons and (much later) binary stars and extrasolar planets. Remember this one!

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Jane Turner [4246] PHY 316 (2003 Spring) Cosmic Distance Ladder

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Jane Turner [4246] PHY 316 (2003 Spring) Special Relativity-what is it? In SR the velocity of light is special, inertial frames are special. Anything moving at the speed of light in one reference frame will move at the speed of light in other inertial frames. SR satisfies Maxwells Equations, which replaced inverse square law electrostatic force by set of equations describing the electromagnetic field SR necessary to get calculation correct where velocities ~ c When velocities << c Newtonian mechanics is an acceptable approximation

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Jane Turner [4246] PHY 316 (2003 Spring) Summary of Formulae Relativistic Doppler z + 1 = √[ (1+v/c)/(1-v/c) ] Relativistic Addition of Velocities length moving =length rest [1-(v 2 /c 2 )] Lorentz Contraction time moving = time rest [1-(v 2 /c 2 )]Time Dilation Lorentz Factor or Mass mass moving =mass rest / [1-(v 2 /c 2 )]

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Jane Turner [4246] PHY 316 (2003 Spring) Example Mass mass moving =mass rest / [1-(v 2 /c 2 )] An object has a mass 1g at rest. What is its mass when traveling at v=0.9999c? mass moving =1g/ [1-(0.9999c 2 /c 2 )] =70.71g !!

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Jane Turner [4246] PHY 316 (2003 Spring) SPACE-TIME DIAGRAMS “Light Cone” light beam follows a world line ct=x, using x versus ct - this is a line at 45 0 object traveling at v 45 0

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Jane Turner [4246] PHY 316 (2003 Spring) SPACE-TIME DIAGRAMS “Light Cone” Inertial Observers Accelerated Observer

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Jane Turner [4246] PHY 316 (2003 Spring) SPACE-TIME DIAGRAMS “Light Cone”

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Jane Turner [4246] PHY 316 (2003 Spring) Special vs General Relativity

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Jane Turner [4246] PHY 316 (2003 Spring) General Relativity General Relativity is a geometrical theory concerning the curvature of Spacetime Gravity is the manifestation of the curvature of Spacetime Gravity is no longer described by a gravitational "field" /”force” but is a manifestation of the distortion of spacetime Matter curves spacetime; the geometry of spacetime determines how matter moves Energy and Mass are equivalent Light is energy, and in general relativity energy is affected by gravity just as mass is

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Jane Turner [4246] PHY 316 (2003 Spring) The Equivalence Principle the effects of gravity are exactly equivalent to the effects of acceleration thus you cannot tell the difference between being in a closed room on Earth and one accelerating through space at 1g any experiments performed would produce the same results in both cases

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Jane Turner [4246] PHY 316 (2003 Spring) Geometries Our homogeneous & isotropic universe can have one of 3 types of geometry

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Jane Turner [4246] PHY 316 (2003 Spring) “Straight Lines” in Curved Spacetime Can examine the geometry of spacetime by looking at the orbits of bodies around large masses - Earths motion around the Sun, not under the force of gravity but following the straightest possible path in curved spacetime (curved due to Suns large mass

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Jane Turner [4246] PHY 316 (2003 Spring) The Metric Equation A metric is the "measure" of the distance between points in a geometry For close points r 2 = f x 2 + 2g x y + h y 2 - metric equation so for any 2 points sum the small steps along the path- integrate! A general spacetime metric is s 2 = c 2 t 2 - c t x- x 2 - for coordinate x , , depend on the geometry Einstein took the spacetime metric, homogeneity, isotropy, local flatness absoluteness of speed of light. Machs idea and the reduction to Newtonian solutions for small gravity ->

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Jane Turner [4246] PHY 316 (2003 Spring) One-line description of the Universe G =8 GT c 4 G, T tensors describing curvature of spacetime & distribution of mass/energy G constant of gravitation labels for the space & time components This one form represents ten eqns! geometry = matter + energy

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Jane Turner [4246] PHY 316 (2003 Spring) GR - Gravitational Redshift We discussed how gravity affects the energy of light Light traveling up ‘against’ gravity loses energy, ie the frequency gets longer (larger) GRAVITATIONAL REDSHIFT! Time between peaks increases -time passes more slowly under strong gravity GRAVITATIONAL TIME DILATION

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Jane Turner [4246] PHY 316 (2003 Spring) Redshift Review Doppler redshift Cosmological redshift Gravitational redshift

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Jane Turner [4246] PHY 316 (2003 Spring) GR Tests: Light Bending Eddington’s measurements of star positions during eclipse of 1919 were found to agree with GR, Einstein rose to the status of a celebrity

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Jane Turner [4246] PHY 316 (2003 Spring) GR-Light Bending Light bending can be most dramatic when a distant galaxy lies behind a very massive object (another galaxy, cluster, or BH) Spacetime curvature from the intervening object can alter different light paths so they in fact converge at Earth - grossly distorting the appearance of the background object

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Jane Turner [4246] PHY 316 (2003 Spring) GR Tests: Planetary orbits GR predicts the orbits of planets to be slightly different to Newtonian physics Long been know there was a deviation of Mercurys orbit vs Newtonian-prediction Einstein delighted to find GR exactly accounted for the discrepancy

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Jane Turner [4246] PHY 316 (2003 Spring) GR Tests: Gravitational Waves Supernova explosion may cause them Massive binary systems cause them & thus lose energy resulting in orbital decay-decays detected! Taylor & Hulse in 1993 -indirect support of GR Changes in mass distribution/gravitational field which changes with time produces ripples in spacetime-gravitational waves Weak Propagate at the speed of light Should compress & expand objects they pass by

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Jane Turner [4246] PHY 316 (2003 Spring) Tests of GR: Line Profiles

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