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Download Notes - SFA Physics and Astronomy
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Gravity The first quantitative model for gravity was given by Newton. He described the force acting between two objects with masses M and m separated by the distance d and showed Fg GMm d2 It’s important to note that the separation is measured between the centers of mass of the two objects. Although we will not use the equation for problem solving, we can learn some important points about the way gravity operates by studying the form of the equation. Gravity is an example of an inverse square force, i.e., the force depends on the inverse square of the separation. By doubling the separation, the force falls by a factor of four. If the separation shrinks by a factor of three, the force increases nine times. Gravity can never become zero. We can increase the separation and make the force as small as we like, but we can’t make it go away. Every piece of matter in the universe has a gravitational attraction to every other piece of matter. Mass causes gravity. Unlike the electrical force which exists as both an attractive and repulsive force, gravity is only attractive. To account for the two types of electrical force, we require two types of charge (positive and negative). Only one type of mass is required. The constant G must be very small so that objects of ordinary mass have negligible gravitational attraction for each other (I didn’t say zero!). You must accumulate a lot of mass before the gravitational force is appreciable. Gravity is the weakest force in nature. Normal forces give us the sensation of having weight. If we remove the supporting (normal) force, we have the sensation of “apparent weightlessness.” So a skydiver feels weightless before opening the parachute. In the tragic example of an elevator breaking loose, you would experience weightlessness. Certain amusement park rides are designed to remove the supporting force and induce a feeling of weightlessness. Galileo correctly concluded that in free fall, all objects must fall at the same rate (g is a constant). While in orbit our astronauts are constantly falling. Since everything around them is also falling and everything falls at the same rate, they have a continuing experience of weightlessness. An interesting consequence of the inverse square nature of the gravitational force is tides. The major influence on the ocean tides of Earth is our Moon. Various parts of the Earth are different distances away from the Moon. Those parts closer to the Moon experience more force than the parts farther away. If we take the gravity acting on the center of the Earth as the average and form force differences, we see that the gravity difference for the side nearest the Moon is directed toward the Moon, but the gravity difference for the side farthest from the Moon is directed away from the Moon. The waters of the oceans respond to this gravity difference and move in the direction of the gravity difference. Thus the sides of the Earth in line with the Moon experience tidal bulges (high tides) while the sides of Earth perpendicular to the Earth-Moon line experience tidal dips (low tides). These tidal variations stay in place with respect to the Earth-Moon line. As the Earth rotates any given position moves into two high tides and two low tides per day (since the Moon orbits the Earth, the actual period is 24h50m for the complete cycle of tides). The Sun contributes a force here as well, but only about half as great as the Moon’s contribution due the distance of the Sun. Greater tides (Spring tides) are experienced when the Sun joins the Earth-Moon line (New Moon or Full Moon). Lesser tides (Neap tides) occur when the Sun is perpendicular to the Earth-Moon line (First or Last Quarter Moon). Other forces can affect the size and exact timing of the tides. Most notable here are friction of the water with the ocean bottom and the shape of the continents. Since ocean tides are obviously present, Earth must also experience atmospheric and crustal tides. The tides of the atmosphere are larger than ocean tides and the tides of the crust very much smaller. Crustal tides have been measured for the Moon and Io, a moon of Jupiter. An alternate view of gravity is to give each body that has mass a gravitational “field.” In the construct we use arrows to visualize the field. For an object like the Earth the field lines are straight and radial, directed toward the center of the Earth. Objects coming near the Earth will experience a force directed in the same way as the field lines. The size of the force depends on how close together the field lines are. Clearly, for the field described for Earth, the arrows are closer together nearer the surface. The concept of a gravitational field allows us to compute the gravity inside the planet. The size of the gravitational field depends only on the mass between you and the center of the Earth. As you move toward the center, the amount of inward mass decreases and so does the gravitational field. At the center of the Earth, the field would have to be zero. The understanding of black holes requires general relativity, a new theory of gravity due to Einstein. Here the universe is viewed as four-dimensional, the three spatial dimensions and time. A universe with no mass has no distortions in space-time. Mass distorts space-time. For example the space-time around the Sun is distorted by the presence of the Sun. Planets are moving in the straightest possible paths given the distorted space-time in which they move. The predicted orbital shapes are ellipses, just like in Newtonian physics. New theories must be able to predict all of the known facts as well as make new predictions to be tested. General relativity does each. The Advance of Mercury's Perihelion - this is a long-standing problem in which the orbit of Mercury does not exactly close onto itself. The point of closest approach to the Sun is slowly moving (advancing) in space. Newtonian physics has no explanation for this effect, but General Relativity predicted it exactly. The Bending of Starlight - gravity affects light just as it affects objects having mass. When light from a star passes close to the Sun, its trajectory is bent. We see the star in a slightly different position than we would if the Sun were not there. This prediction was confirmed in the solar eclipse of 1919. If we could view the vicinity of a collapsing star, as it becomes a black hole, we would notice that the gravity from the star is increasing as the collapse proceeds. At first the gravity is weak and the light is bent very little. As the collapse proceeds the bending becomes greater and greater until light is bent so much that it orbits the star. The orbital radius is called the photon sphere. An exit cone may be defined such that light projected inside the cone is bent, but escapes. Light moving just along the edges of the cone orbits the star, and light projected outside the exit cone is bent so much that it falls back onto the star. As the collapse continues gravity gets stronger and the exit cone narrows. General relativity predicts that the exit cone disappears when the escape velocity becomes equal to the speed of light. Nothing can move faster than the speed of light, so when the gravity has increased to make the escape velocity equal to the speed of light, nothing (not even light) can escape. The surface of the collapsing object at this moment is called the event horizon and its radius is the Schwarzschild radius. The object is truly black. The collapse continues inside the event horizon, but observers on the outside can never see it. The object collapses into a mathematical point called the singularity. The Schwarzschild radius is three times the mass of the collapsed object measured in solar masses. For example a three solar mass black hole has an event horizon of radius 9 km. Near a black hole space-time is severely distorted and time itself is affected. As measured by an outside observer, time stops on the event horizon. Because the original object is no longer accessible from our universe, we cannot know very much about a black hole. In fact, theoretically only three things are knowable about a black hole: Mass - gravity can be felt at a distance Charge - electric forces act at a distance Rotation - material may corotate with the black hole Black holes are frustrating to astronomers - they don't emit any light. Perhaps the only way of finding a black hole is if one formed in a binary star system. If the black hole and the normal star are close enough, material will be pulled off of the normal star and an accretion disk will form. As the material in the disk moves closer to the event horizon of the black hole, it heats up. Just before dipping within the event horizon and disappearing from the universe, the disk gas is hot enough (several million K) to emit Xrays. We may be able to find black holes by looking for X-ray sources. So far several promising candidates have been found, Cygnus X-1 being the best known. Edwin Hubble made a major contribution to the understanding of the structure of the whole universe. He measured the spectra of distance galaxies. He used independent means to determine the distances to the galaxies. Two methods he used were to look at supergiant elliptical galaxies that tend to dominate clusters of galaxies. These galaxies, he found, were about the same brightness and size regardless of where they appeared in the universe. When they appear fainter or smaller, it was because they were farther away. Upon taking the spectra of these galaxies, Hubble found that the each one was redshifted - the galaxies were moving away from us. Even more he found that the more distant galaxies were moving away faster. The information was presented in the form of a graph of recessional velocity vs. distance which we call Hubble's Law. The conclusion we draw from Hubble's Law is that the universe is expanding. Using the graph we can get the distance of anything in the universe as follows: The use of Hubble's Law does rely on the assumption that the redshifts are caused by the universal expansion. The relationship, however, would not be nearly so good if this conclusion were not the fact. Hubble's Law stands today as one of the pillars of Modern Cosmology. The question still remains, however, as to how the universe began. Two cosmologies developed from Hubble's work: The Big Bang Cosmology assumes that the expansion can be drawn back in time to the time when the universe occupied a point. Some process began the expansion (the Big Bang), which continues to this day. Based on the presently observed rate of expansion, we can judge that the universe is 12- 15 billion years old. The Steady State Cosmology assumes that the universe always looks more or less the same. This idea accounts for the expansion of the universe by assuming that matter is continually created to fill in the voids left by the expansion. The amount of material required would be entirely too small to be observed. Prior to the mid-1960's observations could not distinguish between these two cosmologies. Arno Penzias and Robert Wilson at Bell Labs changed all that with their discovery that the universe is emitting blackbody radiation indicative of a blackbody at 3 K. At the same time the Big Bang Cosmologists had determined that the temperature of the Big Bang was sufficient to make Hydrogen and Helium only and, therefore, was limited. Starting with that temperature and expanding the universe for 12 - 15 billion years at the observed rate yielded a universe that was, by now, much cooler. Recall that ideal gases cool when expanded. The present temperature of the Big Bang is 3 K, in exact agreement with the observed microwave background radiation discovered by Wilson and Penzias. Everyone now agrees that the universe began as a Big Bang. But many variations of the basic theory still exist to be sorted out. The 3 K background radiation is another of the pillars of cosmology today.