Strand K. Astrophysics Unit 1. The Expanding Universe Contents Page The Doppler Effect 2 Redshift 5 The Expanding Universe and the Big Bang 8
K.1.1. The Doppler Effect When an ambulance or a police car passes by, the pitch of the siren noticeably changes. The same is true when a train or a loud motorcycle passes by, or when you are watching formula 1 racing on T.V. and the cars pass the camera. This change in pitch is the Doppler effect, and occurs when a source of sound (or light in the case of the relativistic Doppler effect) is moving relative to the observer. If the source is moving toward the observer a higher pitch (shorter wavelength) sound is heard. If the source is moving away from the observer a lower pitch (longer wavelength) sound is heard. In order to understand the Doppler effect, t2 consider Figure K.1.1.1. Here we see a t3 source of sound waves moving toward an t4 observer with a velocity v. At a time t1 the source v source emits a spherical sound wave that emanates from the source (red circle). At time t2 the source has moved to a new position, and emits a second wavefront v (blue circle), whilst the previous wave continues to expand about the point it was emitted. The source continues to move, and at a time t3 emits a third wavefront (red circle) centered at that point in space whilst the two previously emitted sound v waves continue to expand about the point which they were emitted. Thus, although the sound waves emitted are of constant wavelength, due to the velocity of the source (emitting sound waves at different spatial points that then expand about that point), the waves become compressed in front of and drawn out behind the source. As such the frequency of the sound waves v (peaks past a point per second) experienced with the source travelling toward the observer is higher than the frequency observed if the source were travelling away from the observer. This change in frequency (pitch) experienced is the Doppler effect, and is utilised in many technological applications including police radar speed guns. The faster the source, the greater the difference in wavelength relative to the observer. In front of the source with the source travelling toward the observer, the wavelength relative to the observer is; t1 observer λλ tttttttttttt = vv vv ss ff ss 2
and with the source travelling away from the observer λλ aaaaaaaa = vv + vv ss ff ss where v = the speed of the sound wave in air, vs = the velocity of the source and fs is the frequency of the source. Worked Example A police car s siren emits sound waves at a frequency of 300Hz. The speed of sound in air is 340ms -1. Calculate; (a) The wavelength of the sound waves the car is stationary (b) The wavelength and frequency of the sound waves as experienced by an observer with the police car travelling toward them at 30ms -1 (c) The wavelength and frequency of the sound waves as experienced by the observer when the police car has passed, and is then travelling away from them at 30ms -1 Answer For the stationary police car, the velocity of the sound wave is 340m/s and the frequency is 300Hz. Since v=fλ, λλ = vv ff = 340mmss 1 300HHHH = 1.13mm For the police car moving toward the observer, the velocity of the source vs = 30ms -1 and the wavelength relative to the observer is given by λλ tttttttttttt = vv vv ss ff ss = 340mmss 1 30mmss 1 300HHHH = 1.03mm and relative frequency is ff = vv λλ = 340mmss1 1.03mm = 330HHHH 3
For the police car moving away, λλ aaaaaaaa = vv + vv ss ff ss = 340mmss 1 + 30mmss 1 300HHHH = 1.23mm and relative frequency is ff = vv λλ = 340mmss1 1.23mm = 276.4HHHH The Doppler effect also occurs for light, however the relativistic arguments required to express the Doppler effect in terms of frequency are beyond the scope of this course. Instead it is sufficient to understand that, just like sound waves that travel much slower than the speed of light, the wavelength of light relative to an observer with the source moving toward the observer will be shorter and the frequency will be higher. The wavelength will be longer and frequency lower if the source of light is moving away from the observer. Exercise K.1.1. 1. Sketch a plan view schematic diagram of a police car moving away from a stationary observer clearly showing the sound waves being emitted by the car s siren. 2. Calculate the wavelength of the sound from the police car siren when the car is stationary given that the velocity of sound in air is 340ms -1 and the frequency of the siren is 250Hz. 3. The police car then travels at a constant velocity of 20ms -1. Calculate the wavelength and frequency of the sound waves relative to an observer behind the police car (car travelling away from observer). 3. Calculate the frequency and wavelength of the sound waves as observed by a person standing in front of the police car. 4. Now calculate the frequency and wavelength of the sound as experienced by a person in front of the police car (car travelling toward observer). Challenge Question 5. A bat uses sound waves at a frequency of 20kHz to locate a moth. The speed of sound in air is 340 m/s. If the bat flies directly toward the moth at 2m/s, calculate the wavelength and frequency of the sound as experienced by the moth (assuming the moth is stationary). What would the frequency be if the moth was 4
flying toward the bat at 2m/s? (Assume the bat is still moving toward the moth at 2m/s). K.1.2. Redshift When a photon is incident upon an electron orbiting an atom, the electron may absorb the photon, becoming excited (energetically). There are only certain (discrete) excited states available to the electron, and the energy of the photon absorbed must exactly correspond to the difference in energy between the state that the electron originally occupies and the new excited state. Since photon energy is given by EE = hff = hcc/λλ where E is the photon energy, f is the frequency, λ the wavelength of the photon, and h and c are Plank s constant and the speed of light respectively, we can see that since only certain energies are absorbed by an electron, only certain wavelengths of photons will be absorbed. Worked Example An electron with ground state energy of 2 10-19 J absorbs a photon, moving to an excited energy state of 6.42 10-19 J. Calculate the wavelength of the photon absorbed given that h = 6.63 10-34 Js -1 and c = 3 10 8 ms -1. Answer The difference in electron energy is; EE 2 EE 1 = (6.42 2) 10 19 JJ = 4.42 10 19 JJ Rearranging the photon energy equation in terms of wavelength we obtain λλ = hcc EE = 6.63 10 34 3 10 8 4.42 10 19 = 450 10 9 mm = 450nnnn (bbbbbbbb llllllhtt) Each element in the periodic table has a different number of protons in the nucleus and a different electron configuration. Thus the atoms of each element have different allowed energy levels and absorb different wavelengths. The wavelengths that a particular element will absorb is known as the absorption spectrum, and acts Hydrogen Increasing Wavelength 400nm Helium Figure K.1.2.1 750nm 5
as a fingerprint for that element. When light is emitted from an element or passed through an element in gas state, the absorption lines are be visible in the spectrum as dark lines due to the absence of the absorbed wavelength (Figure K.1.2.1.). The absorption pattern can then be compared to patterns obtained from pure elements, allowing substances within the source of the light to be identified. This branch of physics is known as spectroscopy and plays a big role in astrophysics, since the absorption spectrum of light from a distant star may be analysed, and the chemical composition of the star identified. We know that our own Sun contains Spectral lines of Helium helium, since astronomers analysed the spectrum produced by sunlight and (a) found the characteristic spectral lines of helium within it, as shown by Figure K.2.1.2(a). However, when astronomers (b) began to analyse the light from distant stars and galaxies, they were able to determine the characteristic absorption line spectra from common elements 400nm Figure K.1.2.2. such as hydrogen or helium, but all the lines were shifted slightly toward the red end of the spectrum as seen by comparing Figure K.1.2.1 (a) and (b). Since the absorption lines were red-shifted, they appeared at slightly longer wavelengths than expected. Further, it was found that all distant stars and galaxies (far enough away that the gravitational attraction from our own galaxy is negligible) exhibited red-shift, and the further away the galaxy, the greater the shift of the spectral lines toward longer wavelengths. 750nm The shift of the absorption lines toward the red end of the spectrum could only be explained by the Doppler Effect, and since a shift to longer wavelengths is associated with a source moving away from the observer, these redshift observations proved that distant stars and galaxies were moving away from us. Further, light from a stellar object moving toward the observer would, due to the Doppler Effect, be shifted toward the blue end of the spectrum and hence exhibit blue-shift. In the 1920 s American physicist Edwin Hubble collected redshift data on 46 distant galaxies and proved that the further away the galaxy was, the faster it was moving away from us. The only conclusions Hubble could make after these observations were as follows: The further away the galaxy, the bigger the red-shift (the faster it is moving away (greater recessional velocity) All distant galaxies are moving away from us (receding) All the distant galaxies are moving away from each other and from us. Therefore the whole Universe is expanding 6
Due to relativistic effects of the high speeds of receding galaxies the Doppler equation used for sound calculations cannot be used to calculate the redshift of stars. Instead for slow moving stars and galaxies, the redshift ratio z may be calculated. The ratio takes into account the difference in the wavelength of the absorption / transmission line observed from the moving source (λobserved) and the wavelength from a stationary source of the absorption line (λrest) here on Earth. Since z is a ratio, it has no units. The red shift ratio z is given by; zz = λλ oooooooooooooooo λλ rrrrrrrr λλ rrrrrrrr where v = velocity of the moving source = vv cc Exercise K.1.2. 1. State whether the light from the following will be blue-shifted or red-shifted; a. The light from a distant star moving away from us b. The arms of a spiral galaxy that is rotating such that the i. Right arm is moving away from us ii. The left arm is moving toward us c. A star that is moving toward Earth 2. Galaxy A has a greater red-shift than Galaxy B. Which galaxy is further away from us? Justify your answer. 3. A student states that we could not possibly know that the Universe is expanding. Give a brief explanation of the evidence for an expanding Universe. 4. The table below gives the red-shift of different galaxies in nanometers. Place the galaxies in order of distance from Earth. Challenge Question Galaxy Red-shift (nm) M60 18.8 M99 32.1 NGC2366 1.3 NGC2976 0.04 5. The spectrum from a distant object in the night sky is analysed and is found to contain helium. One of the absorption lines occurs at 700nm. Calculate the 7
corresponding gap in energy between the two levels in helium responsible for this absorption line (assume h = 6.626 10-34 m 2 kg/s and c = 3 10 8 m/s). K.1.3. The Expanding Universe and the Big Bang Hubble s discovery that all distant galaxies are red-shifted (moving away from us) and the further away they are the faster the recessional velocity, leads us to the conclusion that the universe is expanding. This can be modeled one dimensionally by starting with a line of equally spaced objects as shown in Figure K.1.3. If we increase the separation between all objects in the line by the same amount in a time interval t, then relative to the object in the middle (object 3) objects 2 and 4 have moved 1 unit whereas objects 1 and 5 have moved 2 units. Since the objects furthest away have moved a greater distance in the time interval t, they must have a greater recessional velocity. Position at t = 0 1 2 3 4 5 Position at t = 1 1 2 3 4 5 Figure K.1.3.1 In order to model the expansion with all objects moving away from each other in two dimensions, one could imagine drawing regularly spaced dots on the surface of a balloon and blowing it up. As the balloon its self expands, all dots on its surface move away from each other. This type of expansion with all objects moving away from each other tells us that just like the balloon and dots, it is space its self (the balloon) that is expanding and not just the objects within it. Since it was discovered that the universe was expanding, scientists have asked the question what will happen to the universe in the end?, to which there are three possible answers. Gravity acts on every single mass in the universe and is always attractive. Therefore gravity, if strong enough, could slow the rate of expansion of the universe or even reverse it. Since the force of gravity is directly proportional to mass, the ultimate fate of the universe depends upon the amount of mass per unit volume it contains (density). Static Universe A static universe is one for which the universe contains a critical density, making the force of gravity exactly strong enough slow the rate of expansion of the universe to zero, and everything becomes stationary as shown by Figure K.1.3.2. This is an unlikely scenario because the law of gravity is not consistent 8
with a static universe. It acts over an infinite range and is always attractive, therefore if the universe contained a mass equal to the critical mass, the expanding universe would be brought to a halt, but gravity would continue to act on all the mass of the universe causing a contraction. Figure K.1.3.2 The Big Yawn If the Universe contains less mass than the critical density, the expansion might be slowed but gravity would not be strong enough to stop the expansion. The Universe would continue to expand forever, cooling as it becomes less dense. Size of Universe The Big Crunch If the universe contains a greater density than the critical density, gravity will be strong enough to stop the expansion of the universe and then reverse it. All objects in the universe will then begin to get closer to each other, ending in a big crunch! Static Universe Time The Big Bang theory is the currently accepted and simplest cosmological model of the universe, describing how the universe expanded from a single infinitely high energy and infinitely high-density point in space. Although the model explains most cosmological observations and is supported with evidence, it is important to remember that it is a theory and is still debated, with many scientists preferring alternative theories arising from string theory and quantum mechanics. When Hubble observed that all distant galaxies were moving away from us at a rate that increased with distance, and that our vantage point was not the center of the universe, the only explanation remaining was that all galaxies are moving away from all other galaxies, i.e. the universe (and space itself) is expanding. Hence yesterday, distant galaxies were closer to us than they are today, and were even closer the day before yesterday. If we reverse time, looking back towards the early universe, then all the galaxies and astronomical bodies must have been much closer together. If we continue to reverse time, looking back to the start of the universe (some 13.7 billion years ago) then the expansion implies that all the mass in the universe must have at the start been together at a single point in space, at infinite density and infinite temperature. We call this point the singularity and assume that the expansion started with a very big bang. Further evidence for the big bang theory comes in the form of cosmic microwave background radiation (CMBR). When scientists began to scrutinise the big bang theory after Hubble s observations, they quickly realised that if the universe was created in a big bang then high energy radiation would have been released. As space expanded, this radiation would also expand, becoming lower energy and longer wavelength over time, and would also permeate throughout space, in all 9
directions. They were even able to show that this background radiation left over from the big bang would by now have a wavelength in the microwave regime. In 1965 two radio astronomers named Arno Penzias and Robert Wilson who were working at Bell Telephone Lab in New Jersey USA accidently discovered CMBR as background noise in one of their experiments, and as predicted it seemed to come from every direction in space, although we now know that it is not as evenly distributed as scientists first thought. Amazingly, this left over heat energy is in part the static that can be heard on a de-tuned radio or the randomly appearing white dots on an analogue television set with no signal and at present, can only be explained by the big bang. Exercise K.1.3. 1. Hubble observed that the further away the galaxy, the greater the redshift. Explain how this observation shows that the universe is expanding. 2. What is meant by the critical density of the universe? 3. Describe what will ultimately happen to the universe if it s density is; a) greater than the critical density b) less that the critical density 4. What is meant by a singularity in the context of the big bang theory? Challenge Question 5. Explain the existence of cosmic microwave background radiation and how it supports the big bang theory. 10