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Weekly Problems Physics 121 November 30, 2009 The story so far: In the 1920s, Edwin Hubble measured the distances (D) and velocities of many galaxies. The results of Hubble’s study were remarkable: • Except for a few nearby galaxies, all galaxies are moving away from us • The farther away a galaxy is, the faster it moves: Hubble’s law: v = H0 D H0=71(km/s)/Mpc Hubble concluded from this that space itself was expanding. Space is expanding outward, so that the distances between galaxies gets larger and larger over time. Today’s class: • Big Bang • Some wave basics we need to know • Redshift • Some assumptions underlying cosmology theory Hubble Law Examples The Big Bang Some galaxy is known to have distance 10 Megaparsecs. Recall that 1 Megaparsec The universe is expanding as we move forward in time. is roughly the size of our “local group” of galaxies, so this puts the galaxy well outside our local group. Imagine going backward in time. • According the the Hubble law, how fast is this galaxy going, in km/s? The universe was smaller and smaller, and denser and denser, • How fast is it going in special relativity units? the further we go back.. We call the starting point of this process the “big bang.” Because velocities are easier to measure than distances, the Hubble law is more often used to estimate distances when velocity is known. • If a galaxy is moving away from us at v=0.1c, how far away is it? 1 The Big Bang Wave Basics: Period, Frequency, Wavelength We have no idea what happened before the big bang, or how to even ask that question. We have no idea why the big bang happened. However, there is very strong evidence that it did, indeed, take place: • The Hubble law tells us that the Universe is rapidly expanding λ • Nuclear reactions in the early stages of the big bang, when the Universe was hotter and denser than the interior of a star, form Helium (and other elements) at This clock “ticks” Period T more-or-less predictable rates. The amount of Helium now seen in the universe with period T. At agrees with expectations from the big bang model. each tick of the clock, • The “cosmic microwave background” is a leftover remnant of photons that were the flashlight emits a 1 created in the hot Universe shortly after the big bang. pulse of light. The Frequency f f = pulses travel to the • As we “look back in time” at distant galaxies, we see fewer and fewer atoms of T elements with high atomic numbers. These elements are only formed in stars, right at the speed of and their relative absence billions of years ago is evidence that the age of the light, c. λ Universe is billions of years. Wavelength λ c = # c = "f The wavelength, λ, is T • As we “look back in time” at clusters of galaxies, they are less and less clustered ! together. Again, this is evidence that the Universe has evolved from a relatively the distance between uniform “soup” of particles into its present state where matter is concentrated pulses. m km 8 5 into small, dense galaxies. Conventional units, c=3×10 s =3×10 s ! ! ! Wave Basics: Photon Energy Wave Example I’ve been doing my best to avoid this formula, but it keeps showing up in the most Hydrogen atoms can be ionized (electron unbound from the nucleus) by absorbing a unexpected places (William Phillips’s talk; Exam #5), so perhaps we should cover it for photon with energy 13.6 eV or greater. real. The energy and frequency of a photon are related by the following formulas. The frequency is the rate at which electromagnetic fields change direction as the photon is • What is the frequency of such a photon? generated or as it passes through space. This is a quantum mechanical observational fact, • What is the wavelength of such a photon? not something we can derive from other more basic principles. • What type of electromagnetic radiation is it? Energy E=hf 2 -34 J -34 kgm Planck’s Constant h=6.63 10 = 6.63 10 × Hz × s eV -15 h=4.14×10 Hz ! ! ! 2 Redshift Redshift and Velocity Emitted Spectrum Because we will usually be working in conventional velocity units (not special relativity units), we will now explicitly include ‘c’ in the Doppler shift formulas. ultraviolet infrared v 400 nm 500 nm 600 nm 700 nm λ − λ 1+ v Z = received emitted = c −1 ≈ λ − v c Observed Spectrum emitted 1 c approximation ultraviolet infrared relativistic Doppler shift for small velocities 400 nm 500 nm 600 nm 700 nm � ( Z +1)2 −1 � v = c � 2 � � (Z +1) +1 � λreceived − λemitted Definition of Redshift: Z = λ emitted Example Example This figure shows a spectrum from a recent study of distant galaxies The spectral lines • What range of redshifts did Edwin Hubble observe? include three hydrogen transitions, with the following emitted wavelengths: Hb (n=4→n=2) 4862Å Hg (n=5→n=2) 4341Å Hd (n=6→n=2) 4102 Å • What is its redshift of this galaxy? • What is its velocity relative to us? • According to the Hubble law, how far away is it? Figure: http://deep.berkeley.edu/gallery. 6500Å 7000Å 7500Å 8000Å 8500Å 9000Å 3 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. 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 what an idealized picture of different). But it is the region. But not in all reference frames. an 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.) 4 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.
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