Extragalactic Astronomy & Cosmology First-Half Review

1. [4246] Physics 316 Extragalactic Astronomy & CosmologyFirst-Half Review

2. Kepler vs Newton

3. 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

4. Law of Universal Gravitation Law of universal gravitation F = GMm/d2 Remember this one! Gravitational force follows an inverse square law- doubling separation between two objects, grav attraction drops x 4 Lecture 3

5. Newtons form of Keplers Third Law Newton also generalized Kepler’s third law as P2=42 R3 /G(M1+M2) Allowed Kepler’s Laws to be applied to moons and (much later) binary stars and extrasolar planets. Remember this one! Lecture 3

6. Cosmic Distance Ladder

7. 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

8. Summary of Formulae Lorentz Factor  or  Time Dilation timemoving = timerest[1-(v2/c2)] Lorentz Contraction lengthmoving =lengthrest [1-(v2/c2)] Mass massmoving =massrest/[1-(v2/c2)] Relativistic Addition of Velocities Relativistic Doppler z + 1 = √[ (1+v/c)/(1-v/c) ]

9. Example An object has a mass 1g at rest. What is its mass when traveling at v=0.9999c? Mass massmoving =massrest/[1-(v2/c2)] massmoving =1g/[1-(0.9999c2/c2)] =70.71g !!

10. SPACE-TIME DIAGRAMS object traveling at v<c has worldline > 450 “Light Cone” light beam follows a world line ct=x , using x versus ct - this is a line at 450

11. SPACE-TIME DIAGRAMS Inertial Observers Accelerated Observer “Light Cone”

12. SPACE-TIME DIAGRAMS “Light Cone”

13. Special vs General Relativity

14. 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

15. 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

16. Geometries Our homogeneous & isotropic universe can have one of 3 types of geometry

17. “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

18. The Metric Equation A metric is the "measure" of the distance between points in a geometry For close points r2= fx2+ 2g x y + hy2 - metric equation so for any 2 points sum the small steps along the path- integrate! A general spacetime metric is s2= ac2t2 -bctx-gx2 - for coordinate x a, b, g 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 ->

19. One-line description of the Universe geometry = matter + energy G=8GT c4 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!

20. 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

21. Redshift Review Doppler redshift Cosmological redshiftGravitational redshift

22. 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

23. 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

24. 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

25. GR Tests: Gravitational Waves 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 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

26. Tests of GR: Line Profiles