Weekly Problems

Physics 121 December 2, 2009 Redshift and Velocity v λ − λ 1 + v Z = r e = c − ≈ v 1 400 nm 500 nm 600 nm 700 nm λ − c e 1 c relativistic Doppler shift approximation 400 nm 500 nm 600 nm 700 nm for small velocities � ( Z +1)2 −1 � v = c � 2 � � ( Z +1) +1� Example • What range of redshifts did Edwin Hubble observe? We are not special! No special places in the universe: Homogeneity More specifically: There is nothing special about our particular place in the universe. On large scales, the universe is homogeneous: every point is like every other. This idea has been influential throughout the history of physics: There are no special places. Earth as center of the solar system? Sun as center of the solar system. The Earth-centered model of the solar This works much better: orbits are simple system requires complicated models to ellipses. We give up the idea of the Earth explain the motion of the planets: as a “special place” in the solar system. Homogenous Not homogenous Inhomogeneous on small scales, but homogenous on large scales On small scales, our universe is inhomogeneous: some places are relatively dense (galaxies, say), other places are nearly empty. If we step back and look over very large scales, 100s of Megaparsecs, say, every place looks pretty much like every other place: for example, the number of galaxies per unit The idea that we are not in a special place (or a special time) is sometimes volume doesn’t vary by much. called the Copernican principle, after Nicolaus Copernicus, who proposed the Sun-centered solar system model. 1 No special directions in the universe: Isotropy On large scales, the universe is isotropic: every direction is like every other. Homogeneity and isotropy are related to one of the core principles of physics: The laws of physics are the same at all times, in all places, in all reference frames, and in all directions. If you a perform an experiment, and then you repeat the exact same experiment in a different place or a different time, you will get the same result. We are now going a step farther. We are saying that not only that the laws of physics the same everywhere, but also that the “stuff” that the universe is made of Not isotropic (vertical, Isotropic, from the point Homogenous and isotropic. is the same in all places and in all directions. On the other hand, the "stuff" is not horizontal directions are of view of the center of the same at all times (the universe was more dense in the past) and is not the same This is an idealized picture of an different). But it is the region. But not in all reference frames. ideal universe on large scales: homogenous on large homogenous: the center very boring! scales. is a special place. On small scales, our universe is non-isotropic: look in one direction and you will see more nearby galaxies than in another direction. On large scales, it is isotropic: you see the same number of distant galaxies no matter which direction you look in. (You will also see nearly the same intensity of microwave background radiation from all directions. We will discuss this relic of the big bang on in another class.) A related (incorrect) idea: Steady State Universe Steady State Universe, continued The idea of a steady state universe is that no point in time is special. Einstein solved the problem of the buildup of density inhomogenities by introducing the cosmological constant. This is an extra term in the equations of general relativity This contradicts the observed fact of the big bang, since the moment at which the big bang which acts like a repulse force. happened is surely a special point in time. So the “steady state” idea is wrong. Nevertheless, the idea of a steady state universe played an important role in defining how Here’s his idea: we talk about the universe today. • Because of the cosmological constant, every point in space repels every other point Einstein developed general relativity before Hubble’s discovery of the expanding in space. Universe. Einstein thought that the universe was both homogenous (no special points in • At the same, time because of mass distributed throughout the universe, every point space) and steady state (no special points in time). attracts every other point by the gravitational force. There was a problem with this picture, which we can understand in Newtonian terms. • These effects exactly balance, so that the universe remains in steady state, with mass Every bit of mass is gravitationally attracted to every other bit of mass. If the universe is (such as galaxies) neither rushing towards each other nor rushing away. absolutely perfectly homogenous (it has uniform mass density), then each bit of mass will be pulled equally in all directions, and will not move. (Which is not a problem.) If The rationale for the cosmological constant was wrong. The universe is not in a steady however, there are areas of the universe which are just slightly denser (more massive) state: as Hubble showed, it is expanding. Upon learning of Hubble’s discovery, than their surroundings, they will tend to pull their surroundings in toward them. This is a Einstein called the cosmological constant “the biggest blunder of my life.” problem: as more mass gets pulled in, the attractive force is larger, and the process snowballs. For an infinitely old universe, this is a very big problem, because it creates Recent astronomical observations have shown that, in fact, a cosmological constant large mass inhomogeneity. seems to be needed after all. (We’ll see why soon.) 2 The Fate of the Universe Back to Hubble’s Strategy All the Galaxies are rushing away from each other. Yet, they are attracted to each other Hubble’s strategy was to measure distances and redshifts (velocities) of distant by gravity. We expect the gravitational force to decelerate the galaxies’ motions. galaxies. This remains a powerful way of studying cosmology. The goal is to measure (If the direction of an object’s acceleration is opposite to the direction of its galaxies at larger and larger distances. motion, the object slows down.) In this deceleration scenario, there are two possibilities: The basic idea remains the same: observe something of known luminosity (power), measure the intensity of its light at the Earth, and use the formula • If the deceleration is strong enough, the galaxies eventually reverse course and start P hurling at one another. The universe ends in a “big crunch” billions of years into the I = 4πr 2 future. Watch out! to find the distance, r. • If the deceleration is not too strong, the Universe keeps expanding forever, but it goes more and more slowly over time. The challenge is to find something of “known luminosity.” Sometimes the term “standard candle” is used to describe such objects. A problem arises: Various astronomical observational programs have been undertaken in order to determine which of these possibilities is the fate of the Universe. To the surprise of • Some types of stars, such as Cepheid variables, are good standard candles. However, these everyone, neither of these possibilities turned out to be correct. stars are not bright enough to be seen in very distant galaxies, so they won’t work. • Galaxies themselves are bright enough to be seen, but there is nothing “standard” about them. Looking at a galaxy, you can’t tell how bright it is supposed to be, so you have no “P” to plug into the I= P/4πr^2 formula. • We need something else! What we need are ....... supernovae! Supernovae In the 1930s, Walter Baade and Fritz Zwicky realized that the collapses of some types of stars would trigger explosions, blowing most or all of the star outward. We Gravitational call these events supernovae. They happen during the death throes of very massive Pressure stars (stars that initially have, say, 8 times more mass than our own Sun). In 1054, Chinese astronomers noted that an extremely bright Gas new star had appeared in the sky: we now know this to have In a star like our sun, been a supernova. Over weeks and months it faded away. p + p 2H + e+ + → νe Pressure outward pressure of hot gas 2H + p → 3He + γ Even now, the remnant of the supernova can be seen. (heated by thermonuclear 86% 14% In fact, it is still expanding outward into space! reactions) balances inward 3 4 7 pressure from gravity. 2H + 3He → He + He → Be + γ Supernova are very rare: there is only one 4He + 2p + γ every hundred years or so in our own 14% 0.02% Eventually the thermonuclear Crab Supernova remnant Galaxy. (The most recent nearby reactions cease—the nuclear 7 – 8 7Be + e– → 7Li + ν Be + e → B + νe supernova was in 1987, in the Large e 8 8 + fuel is “used up”—in which 7Li + p → 2 4He B → Be + e + νe Magellanic cloud, a satellite galaxy of our 8Be → 2 4He case the star collapses. own Milky Way galaxy.) Supernova are very bright and hence can be seen at great distances. Fritz Zwicky 3 Supernovae as Cosmological Probes Because supernovae can be seen very far, they are potentially useful to help us measure distances. But: we need to find “standard candles,” categories of objects all of which have the same intrinsic luminosity. Are supernovae standard candles? Most are not. The luminosity of a supernova explosion depend on the composition of the star and the circumstances of the explosion. Example of a supernova “light curve” showing intensity versus time measured in four different colors.

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