Slide 1Center of MassSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Mercury: the closest planet to the SunSlide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40The Galactic CenterSlide 42A Black Hole at the Center of Our GalaxySlide 44Slide 45Slide 46Slide 47Slide 48Fate of the collapsed coreSlide 50Slide 51Slide 52Slide 53Center of MassSlide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Orbital Motion in the Milky WaySlide 63Slide 64Slide 65Slide 66Slide 67Slide 68What is dark matter???Slide 70Gravity in Newtonian physics2rmMGF mMCenter of Mass(SLIDESHOW MODE ONLY)Binary systemsIf we can calculate the total mass and measure the mass of a normal star independently, we can find the mass of an unseen companionApplications: • Extrasolar planetsSee http://faculty.physics.tamu.edu/belyanin/math%20and%20cosmos.ppt• Black holes and neutron stars in binary systemsSee http://faculty.physics.tamu.edu/belyanin/SMP2007%20long.pptEscape condition: Kinetic Energy K Gravitational Potential Energy |U|RGMmUmVK ;2/2At threshold: RGMmUmVKesc22RGMVesc2Object Mass Escape velocityCeres (largest asteroid)1021 kg 0.64 km/sThe Moon 7x1022 kg 2.38 km/sThe Earth 6x1024 kg 11.2 km/sJupiter 2x1027 kg 60 km/sThe Sun 2x1030 kg 618 km/sWhat happens with even more massive and dense objects?Escape velocities for some objects222cGMRcRGMVssescCritical (Schwarzschild) radiusBlack holes in Newtonian physicsFirst suggested by Laplace in 1796The result is accidentally correct, but derivation is wrong and picture is wrong. We need general relativity!Newton’s theory is a weak-gravity limit of a more general theory: General RelativityEven in the weak gravity of the Earth and the Sun, there are measurable deviations from Newtonian mechanics and gravitation law!• Advance of Mercury’s perihelion• Bending of light by the Sun’s gravityGeneral Relativity predicts new effects, completely absent in the Newton’s theory: black holes, event horizons, gravitational waves.General RelativityDeveloped in 1907-1915 by A. Einstein in close collaboration with mathematicians: Grossmann, Hilbert, Levi-Civita ... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. Marcel Grossmann David Hilbert Tullio Levi-CivitaAlbert EinsteinGravity is a strange force. It has a unique property:MmR2RmMGF 2RMGmFa All bodies in the same point in space experience the same acceleration! Galileo, about 16002RMmGamgi!!!gimm In 1907, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life , namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence:- ... we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame. Equivalence PrincipleThis means that in the freely-falling elevator cabin you don’t feel any effects of gravity! You and all objects around you experience weightlessness. Vice versa: in outer space you can imitate the effect of gravity by acceleration.MmR1211RMGa Warning: all bodies experience the same acceleration, but only in a small region of space. In another region this acceleration is different. Time flows with a different rate, and paths are bent differently in these two regions. R2222RMGa If gravity can be eliminated or imitated by motion, no special force of gravity is needed! How to explain that in the absence of any force the trajectories are not straight lines?Because space and time are curved by the matter!The force of gravity is actually the acceleration you feel when you move through space-timeSpace-time gets curved by masses. Objects traveling in curved space-time have their paths deflected, as if a force has acted on them.Main idea:“Curvature” of time means that the time flows with a different rate in different points in space"Matter tells spacetime how to bend and spacetime returns the compliment by telling matter how to move."John WheelerAbout 1912 Einstein realized that the geometry of our world should be non-Euclidean.He consulted his friend Grossmann who was able to tell Einstein of the important developments of Riemann, Ricci and Levi-Civita. G.F.B. Riemann(1826-1866) When Planck visited Einstein in 1913 and Einstein told him the present state of his theories Planck said: As an older friend I must advise you against it for in the first place you will not succeed, and even if you succeed no one will believe you.Several versions of Einstein’s GR in 1913-1914 were wrong. Only in November 1915, after correspondence with Levi-Civita and Hilbert, Einstein published a paper with correct equations. Hilbert also published correct equations, in fact 5 days earlier than Einstein. On the 18th November Einstein made a discovery about which he wrote For a few days I was beside myself with joyous excitement . He explained the advance of the perihelion of Mercury with his theory.The advance of the perihelion of Mercury One little speck on the brilliant face of Newton’s theory:Mercury: the closest planet to the SunSunMercuryPerihelion = position closest to the sunAphelion = position furthest away from the sunPerihelion: 46 million km; Aphelion: 70 million kmMercury's perihelion precession: 5600.73 arcseconds per century Newtonian perturbations from other planets: 5557.62 arcseconds per century Remains unexplained: 43 arcseconds/century (Le Verrier 1855)In reality the orbits deviate from elliptical:1 degree = 3600 arcsecondsUrbain Le Verrier 1811-1877Predicted the presence and position of Neptunefrom irregularities in Uranus’s orbitNeptune was found in 1846 exactly at the predicted position In 1855 Le Verrier found that the perihelion of Mercury advanced slightly more than the Newtonian theory
View Full Document
Unlocking...