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Name ……………………………………………………… Advancing Physics A2 John Mascall The King’s School, Ely Chapter 12 Our place in the universe (Parts 1 and 3) Student Notes August 2009 Assessable learning outcomes for all of Chapter 12 Candidates should demonstrate evidence of: 1. knowledge and understanding of phenomena, concepts and relationships by describing: (i) the use of radar-type measurements to determine distances within the solar system; how distance is measured and defined in units of time, assuming the relativistic principle of the invariance of the speed of light; (ii) effect of relativistic time dilation using the relativistic factor 1 1 v2 / c2 ; (iii) the measurement of relative velocities by radar observation; (simple arguments using two successive pulses are sufficient); (iv) evidence of a ‘hot big bang’ origin of the universe from: • cosmological red-shifts (Hubble’s law); • cosmological microwave background; 2. scientific communication and comprehension of the language and representations of physics, by sketching and interpreting: (i) logarithmic scales of magnitudes of quantities: distance, size, mass, energy, power, brightness; 3. quantitative and mathematical skills, knowledge and understanding by making calculations and estimates involving: (i) distances and ages of astronomical objects; (ii) distances and relative velocities from radar-type measurements. A Revision Checklist for Chapter 12 can be found on the Advancing Physics CD-ROM. Page | 2 Ch 12.1 Observing the Universe Learning outcomes Astronomical distances in the Solar System can be measured by radar Distances to nearby stars can be found by parallax Some larger distances are estimated by the apparent brightness of ‘standard candles’, e.g. Cepheid variables and Type 1a supernovae The cosmological distance scale is still subject to uncertainty For v much less than c, velocities of astronomical objects can be established by the Doppler shift with λ v λ c Masses of astronomical objects can be found from the velocities of objects orbiting them at known distances or in a known time. Spectra of distant objects over a wide range of wavelengths provide knowledge of their chemical composition Looking out on the Universe We start with a discussion around what you already know about cosmology and ask the question, 'And how do you know?'. Activity 10E Experiment 'What do you know about cosmology?' Many people have a great interest in cosmology – and already know a lot about it. But what they know may be disorganised - and self contradictory. This activity will help you find out how much you already know about this topic. We follow this by taking a look at images of the Universe at a wide range of wavelengths: This first computer image (Display Material 20S Computer Screen 'The Milky Way in infrared') is given as an example. All can be found on the CD-ROM) and on the PowerPoint ‘Looking in on the Universe’. Display Material 20S Display Material 40S Display Material 50S Display Material 60S Display Material 70S Computer Screen 'The Milky Way in infrared' Computer Screen 'Eta Carinae in visible light' Computer Screen 'An ultraviolet image of the Sun' Computer Screen 'Images of supernova remnant Cassiopeia A' Computer Screen 'X-ray and gamma ray images' Page | 3 The following OHTs help to summarise what we know about astronomical distances. Display Material 10O OHT 'Distances in light travel time' Page | 4 Distances in light travel-time 1027 m 1000 million light-years 1 million light-years limit of observable Universe Hubble deep field 1024 m distance to Andromeda galaxy 1021 m size of Milky Way galaxy 10 18 m 1 light-year 10 15 m next-nearest star Alpha Centauri Sun to Pluto 1 light-hour 10 12 m Sun to Earth Sp eed o f ligh t 1 light-second 109 m 106 m 1 lightmicrosecond 103 m Earth to M oon 300 000 kilometres per second 300 km per millisecond 300 m per microsecond 300 mm per nanosecond Ligh t travel-tim e One hour by jet plane ten m inute walk 1 1 1 1 light- second = 300 000 km light- millisecond = 300 km light- micros econd = 300 m light- nanosecond = 300 mm Tim e 1 year = 31.5 million seconds 1 lightnanosecond 1m L ig ht-year 1 light-year = 9.4 10 15 m 16 = 10 m approximately Page | 5 Display Material 30O OHT 'The ladder of astronomical distances' Ladder of astronomical distances 10 10 Red-shift Assume that the speed of recession as measured by wavelength shift is proportional to distance Brightest galaxy Assume that brightest galaxies in clusters are all equally bright 10 9 Super nova Type 1a supernovae all have the same absoute brightness Coma cluster 10 8 Virgo cluster of galaxies 10 7 Tully-F isher F aster rotating galaxies have greater mass and are brighter M31 Andromeda 10 6 Magellanic clouds Blue supergiants Assume that the brightest star in a galaxy is as bright as the brightest in another 10 5 10 4 Cepheid variables T hese very bright pulsing stars can be seen at great distances . T he bigger thay are the brighter they s hine and the slower they pulsate. 10 3 10 2 10 1 Colour-luminosity The hotter a star the brighter its light, and the brighter it shines. If the type of star can be identified there is a known relationship between colour and brightness. Distance then found comparing actual with apparent brightness Parallax Shift in apparent position as Earth moves in orbit round Sun. Rec ently improv ed by using satelite Hippar cos: now overlaps Cepheid scale. Baseline all distances based on measurement of solar system, previously using parallax, today using radar Page | 6 Above are some techniques for measuring distance, arranged to show how they overlap, allowing comparisons between the techniques to be made. The following experiments give you some idea of how to find out about objects that cannot be reached. They provide useful background information but are not essential. Activity 20E Experiment 'Range finding and parallax' In the activity you will learn about trigonometric parallax and how the distances to nearby stars can be measured. Activity 30E Experiment 'A homemade reflection spectroscope' You will use this to observe the Sun’s spectrum and to look for absorption lines – missing lines in the spectrum caused by absorption of certain wavelengths by the Sun’s atmosphere. The PowerPoint presentation called ‘Spectra’ shows the three main types of spectra together with an example of the Sun’s spectrum. Activity 40E Experiment 'Brightness and distance' Using the intensity of starlight to measure the distance to the stars is often difficult. You have to assume the brightness of the star, and then use the way the brightness of the star varies as you get further away, perhaps assuming a 1/r2 variation, to produce a distance. Activity 50E Experiment 'Summer Sun remembered’ In this activity you compare energies incident on the face which allows you to make an estimate of the distance of the Sun. Notes: Page | 7 Radar ranging and Doppler shift By using a single pulse of radio waves we can measure how far a distant object is away provided it is not too far. You should appreciate why there is no possibility of using this method to measure the distance to the nearest star beyond the Sun, which is 4.2 lightyears away. If we send two pulses of radio waves a known time interval apart we can measure the relative velocity of a moving object. Again, this method will not work if the object is too far away, but it will work for asteroids passing close to the Earth. Display Material 90O OHT 'Velocities from radar ranging' Radar ranging re lativ e velocity v relative ve locity v asteroid as teroid firs t radar pulse out spe ed of light 8 –1 = 3 10 m s firs t radar puls e returns s ec ond radar s econd radar pu lse o ut pulse returns fir st p u lse seco nd p ulse time betwe en first and sec ond pulses 10 0 s first pulse out first puls e returns sec ond pulse out 0.2 s sec ond puls e re turns 0.22 s 100 s distance out and bac k = 0.2 light-secon ds distance out and bac k = 0.22 light-secon ds distanc e of asteroid = 0.1 light-s econd s = 30 000 k m distanc e of asteroid = 0.11 light-sec onds = 33 000 km time taken = 100 s increase in dis tance 3000 k m = 0.01 light-se conds relative veloc ity v = 3000 km 100 s –1 v = 30 k m s 4 v/c = 0. 01 s/100 s = 10 – The relative velocity of an asteroid is calculated by measuring two outand-back radar pulses Page | 8 For objects that are a very long way away (distant stars and galaxies) the signal would take too long to return and would be too weak to detect. In this case we must make use of the Doppler effect. Atoms in the distant stars emit light at particular wavelengths. If we can identify the elements from which the light is emitted it is possible to measure these wavelengths accurately on Earth. If the stars or galaxies are moving away from us we will detect an increase in wavelength when the light reaches Earth. This is known as a redshift. Display Material 95O OHT 'Non-relativistic Doppler shift' Doppler shift source and receiver both at rest c = speed of light f = frequency T = time for 1 cycle receiver source w avefronts wavelength = c = cT f v + cT+ vT = =1 + cT c v = c wavelength larger by + = cT + vT receiver source moving and receiver at rest velocity v source travels distance vT in each cycle The wavelength increases when the source travels away from the receiver We can illustrate the Doppler effect with the sound from a Formula 1 car as it races past or with a Doppler ball swung around the head! The Doppler effect is particularly well illustrated in context in the IOP Schools’ Lecture presentation by Pete Edwards called ‘Gravity, Gas and Stardust’. Page | 9 Some or all of the following activities may be used: Activity 110P Presentation 'Changes in velocity and rotation' Also uses File 50L Launchable File 'Doppler seen and heard' Many objects in the Universe spin on their axis or orbit around a more massive body. Typically this will result in components of their velocity, as seen from Earth, varying. Sometimes the radiating atoms on the object will be, on average, moving towards the detector, and sometimes away from the detector. So the frequency of the detected waves will vary. Often this will be the only evidence available. Exactly this observation has been used to claim the first direct sighting of a planet, 55 million light-years away, in December 1999. Here you look at a model of this kind of remote sensing of velocity. Activity 120D Demonstration ‘ Doppler shift using microwaves’ In this experiment, a moving metal sheet is used to reflect microwaves and, because the sheet is moving, the frequency of the reflected waves is shifted. The frequency shift depends on the speed and direction of the movement. Demo Activity 90S Software Based 'The space police' Using File 30L Launchable File 'Radar ranging model' Key point: ∆ = v c where v c See pages 68-69 of the text. Question A spectral line resulting from a transition within hydrogen atoms associated with a quasar is found have wavelength of 475.0 nm when detected on Earth. The same transition from hydrogen in the laboratory produces radiation of wavelength 410.2 nm. (a) Explain why this implies that the quasar is moving away from us. (b) Use the Doppler formula to calculate the velocity of the quasar. Notes: Page | 10 Doppler shifts (used for measuring speeds) have been crucial to detecting rotations of galaxies and other objects. Here the rotation is used to calculate the mass of a candidate black hole. The argument for measuring the mass of black hole is developed on pp 71-72 of the text. Display Material 190O OHT 'Measuring black holes' Measuring the mass of a black hole –1 velocity 209 km s away from us acc eleration g = com panion s tar v2 r ac cele ration meas ures the grav itational field of the black hole acce leratio n g cand idate black hole field g = m as s M radi us r GM r2 = GM r2 v2 r 2 v eloc ity 209 km s –1 mas s from v and r M = v r G towards us Example: mass o f a cand idate black h ole V404 Cygn us V404 Cy gnus emi ts X-ray s perhaps due to m atter from an ordinary star falling into a m ass ive black hole c ompanion w hich it orbits . Dopple r s hifts of light from the ordinary s tar show its ve locity varying by plus or min us 209 k m s–1 over a period of 6.5 d ays . speed v in o rbit from D oppler measurements , ass uming orbital plane is in the lin e of s ight radiu s r o f orb it from time of orbit and speed of star: 1 star travels 2r in 6.5 days at 209 km s – acceleratio n a to ward s b lack h ole acc eleration = v2 r = 9 r = 18.7 10 m (209 10 3 m s–1)2 2 a = 2 .34 m s– 18.7 10 m 9 s am e quantity 1 g = 2.34 N k g – gravitatio nal field g of black gravitational field = acc eleration ma ss M o f b lack ho le from grav itational invers e squarefield g = GM r –1 2 M= 9 gr 2.34 Nkg (18.7 10 m ) = G 6.67 10 – 11 Nm 2 kg – 2 m ass of Sun = 2 10 30 kg 1 v = 2 09 k m s– 2 2 M = 1.2 10 31 kg mas s M = 6 times m ass of Sun The mass of a black hole can be calculated using the velocity of a star orbiting it Page | 11 Evidence of red and blue shifts from the same galaxy is given below: Display Material 130S Computer Screen 'Doppler shifts from part of a galaxy' Images of the central gas disc of galaxy M87, and the spectra of emission line gas from that central disc, showing both red shifts and blue shifts. These show that one side of this gas disc is approaching us whilst the other side is receding. The disc seems to be rotating at a speed of about 550 km s-1 . There is no evidence that the whole galaxy is rotating. Page | 12 Ch 12.2 Special relativity Learning outcomes Motion is relative: 'being 'at rest' means only 'moving with me'. The speed of light c is constant, the same relative to all objects, however they move. The relativistic Doppler shift approximates closely to the non-relativistic value 1 + v / c for v substantially less than c, but increases rapidly as v approaches c. The time dilation factor γ is very close to 1 for v substantially less than c but increases rapidly as v approaches c. We start with a discussion of what is involved in ‘thinking relatively’. In Question 50C Comprehension ‘Thinking relatively’ you have to learn to describe events seen from different points of view, and work out what changes and what stays the same. These questions provide simple exercises in thinking in this way and require no knowledge of the results of the theory of relativity. Answers to each question are provided on a PowerPoint to support class discussion. The following limerick taken from the student text encapsulates the ideas met so far. When Einstein was travelling to Spain, He drove the conductor insane: “It may be a while,” He would ask with a smile, “But when does Madrid reach this train?” We have seen in Chapter 11 that the law of conservation of momentum applies regardless of the speed of a moving observation platform. Travelling at a different constant velocity relative to another object does not change the laws of physics. We must accept another essential idea about light and that is that its speed is the same regardless of the speed of the observer. That speed is now defined as 299 792 458 ms -1. There is good experimental evidence to support the constancy of the speed of light and this is backed up by a coherent set of theoretical arguments. Some evidence is outlined on pages 76-77 of the student text. The two postulates that underpin special relativity are as follows: Postulate 1 The laws of physics are the same for all observers moving at a constant velocity. Postulate 2 The speed of light is the same for all observers. It does not depend on the speed of the source or the speed of the observer. Page | 13 In Section 12.1 we discussed a radar method to measure the distance and speed of an asteroid. We assumed that the radar pulses travelled at a constant speed we now know to be the same constant speed for all electromagnetic waves when travelling in a vacuum. One pulse gave us a single distance measurement and two pulses gave us a change of distance in a known time. In the arguments below we use space-time diagrams to represent radar measurements. Display Material 100O OHT 'Space-time diagrams' Space time diagram s tim e t/s 10 pulse back 4s back tim e t/s 10 radar pulse out and back radar pulse out and bac k pulse back 8 x /c = 4 s 3s back 6 8 6 x /c = 3 s 4s out 4 radar puls e out and back 2 pulse out your w orldline trav elling in time 3s out worldline of objec t not moving relative to you goes vertic ally 0 4 2 pulse out sloping w orldline of object m oving relative to you 0 2 4 distance x/c 6 2 4 dis tance x /c 6 D istance is m easured by reflecting a radar pulse then measuring the time sent and the tim e received back . T he clock us ed travels w ith you and the radar equipm ent (at rest). As sum ptions: speed c is constant s o reflection oc curs halfw ay through the out-and-back time speed c is not affected by the m otion of the distant objec t Radar measures distances using light-travel times The constant speed of light is assumed The key features of a space-time diagram are given on page 77 of the student text. You should make sure you are understand these diagrams before you move on. Questions 1 Why does the worldline for the moving object in the right-hand panel have a greater slope than that for the radar pulse? ………………………………………………………………………………………………… ………………………………………………………………………………………………… Page | 14 2 (a) Using information given on the right-hand panel, calculate how far away from you the moving object is at the moment the radar pulse reaches it. Give your answer in both light-seconds and metres. ………………………………………………………………………………………………… ………………………………………………………………………………………………… (b) How far away is the moving object at the moment the radar pulse returns to you? Choose the most appropriate unit for your answer. ………………………………………………………………………………………………… ………………………………………………………………………………………………… Measuring speed Display Material 110O OHT 'Two-way radar speed measurement' Two-way radar speed m easurm ent (simplified) time t/s 10 worldline of observer pulse 2 back t2 distance at moment of reflection worldline of moving object from pulse travel time 1 = c 2 (t2 – t1 ) 8 from object travel time 1 = v 2 (t2 + t1 ) 6 pulse is reflected 5 4 v t2 – t 1 = c t2 + t 1 1 (t – t1 ) 2 2 1 (t + t1 ) 2 2 2 pulse 2 out t1 t2 1 + v/c = t1 1 – v/c pulse 1 out and back instantly 0 3 ls distance x/c pulse takes 6 s to go and return distance = 3 light-s pulse sent at t = 2 s must have been reflected at t = 5 s 6 ls t2 = 8 s t2 =4 t1 = 2 s t1 v 8s–2 s 3 = = c 8s+2s 5 Speed is measured by comparing the interval between returning pulses with the interval at which they were sent Note that, for simplicity, the object moves past the observer at t = 0 and the first radar pulse is sent out at that time. Clearly it takes negligible time for this pulse to be returned after being sent out. The second pulse can then be used to find out how far the object travels in a certain time. The panel shows that the distance at the moment of reflection can be expressed as c(t2 – t1)/2 and as v(t2 + t1)/2. Page | 15 Questions 1 Show that equating these two distances gives v/c = (t2 – t1)/(t2 + t1) as shown. ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… 2 Re-arrange the equation v/c = (t2 – t1)/(t2 + t1) to show that t2/t1= (1 + v/c)/(1-v/c) as given. ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Doppler effect Remembering that the first pulse went out and returned at t = 0, you should be able to use OHT 110O to convince yourself that: t1 = interval between pulses sent out and t2 = interval between pulses coming back. It therefore follows that interval between pulses coming back = ∆t back = 1 + v/c interval between pulses sent out ∆t out 1- v/c Question Show that ∆t back / ∆t out when v = 0.1 c is 1.22. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… You have shown that ∆t back is 1.22 larger than ∆t out. This stretching of the interval between a pair of pulses is the Doppler effect occurring twice. Firstly, the pulses are being sent to an object moving away from the original source of the pulses. We say that this introduces a Doppler stretching factor k. Secondly, the pulses are returned to the observer from the object that now acts as a moving source. This introduces another Doppler stretching factor k. Page | 16 It follows that the interval between the pulses when they return is given by ∆t back = k × k × ∆t out = k2 ∆t out Question Calculate the value of k in the numerical example given above. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… We can now establish that ∆t back = k2 = 1 + v/c ∆t out 1- v/c The arguments are summarised below. OHT 'Doppler shift – two-way and one-way' Display Material 120O Doppler shift – two-way and one-way time t 10 B A Let the one-way Doppler shift = k Pulses sent from A at intervals t will arrive at B at intervals k t out 8 pulses return to observer at intervals t back = k ktout The same shift k must apply to pulses sent from B to A Therefore: pulses arrive back at A at intervals t back = k kt out 6 pulses arrive at moving object at larger intervals ktout 4 pulses sent out at intervals t out The two-way Doppler shift = k 2 Two-way Doppler shift 1 + v/c k2 = 1 – v/c 2 One-way Doppler shift 1 + v/c k= 1 – v/c 0 2 4 distance x/c 6 The Doppler shift k is the observed quantity that measures the speed of a remote object We are normally interested in the one-way Doppler effect for which k 1 v / c 1 v / c Page | 17 We met k as the stretching factor for the time between a pair of pulses when the object is moving away. This applies to a one-way trip only. The same idea applies to the stretching of wavelength when source and observer are moving apart. This follows since the wavelength is proportional to the time for one cycle. The relativity equations we have just developed now replace the previous equations given for the Doppler effect (see page 68 of the student text). From relativity we have k received 1 v / c sent 1 v / c where λreceived = λ + ∆λ λsent λ The previous expression for the Doppler effect (no reference to relativity) was λ + ∆λ = 1 + v/c λ Whilst these equations look very different, the panel below shows that they give very similar values when v « c. The panel below shows that the relativistic Doppler shift can be written as λ + ∆λ = γ(1 + v/c) λ where 1 1 v2 / c2 Questions 1 Show that γ is only 1.005 when v = 0.1 c. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… 2 Explain why the simpler expression for the Doppler effect is adequate when speeds are lower than 0.1 c. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… Page | 18 Display Material 125O OHT 'Relativistic effects' Relativistic effects Relativistic Doppler shift is: Relativistic and non-relativistic Doppler shifts 7 + = 1+v/c 1–v/c 6 relativ istic Do ppler s hift (1 + v /c ) 5 4 3 identic al for v < < c 2 multiply by 1+v/c =1 1+v/c 1 non- relativ istic Doppler shift 1 + v /c 0 0 0.1 0.2 0 .3 0.4 0.5 v/c 0.6 0.7 0.8 0.9 1.0 0.9 1.0 giving + 1+v/c = (1–v/c)(1+v/c) Relativistic factor 4 = 1 2 2 (1 – v /c ) inc reases towards infin ity as v approac hes c 3 which is is alw ay s > 1 + 1+v/c = (1–v 2/c2) 2 ~~ 1 for all v << c 1 0 0 0.1 0.2 0.3 0.4 0.5 v/c 0.6 0.7 0.8 it is neater to write this as + (1+v/c) = with 1 = 1–v 2/c 2 At low speeds relativistic and non-relativistic results are the same. At speeds near c the relativistic factor increases rapidly The following activities are designed to support the work on the relativistic Doppler effect. Activity 100S Software-based ‘The relativistic Doppler effect’ In this exercise you will explore the relativistic Doppler effect that gives red shifts and blue shifts in the spectra of stars and galaxies according to whether they are moving away from us or towards us. You will be able to turn off the effect of special relativity and see what happens. In this way you will learn some surprising things about time. Activity 50S Software-based ‘Investigating the relativistic Doppler shift’ The theory of relativity gives the relativistic Doppler shift k as k 1 v / c 1 v / c Here you study the way that k varies with the ratio v / c. You can compare the values with the non-relativistic approximation 1 + v / c for a moving source. Page | 19 Time dilation and the relativistic factor γ In the simple Doppler shift equation, the factor 1 + v/c comes about because the source is moving away from the observer stretching out the light waves. The term γ in the relativistic equation γ (1 + v/c) arises because clocks in the systems moving relative to each other cannot agree. A clock moving relative to you runs slower by a factor γ compared with a clock moving at your speed. This is known as time dilation. In the panel below a clock moves sideways to avoid any classic Doppler shift. Any difference between the clocks can only be due to time dilation. The clock used is known as a light-clock consisting of a pair of mirrors between which pulses of light bounce back and forth. Display Material 13O OHT 'The light clock' The light clock sitting beside the clock clock travelling past you at speed v Pythagoras’ theorem: (c)2 = (c)2 – x 2 2 = t 2 – (x/c) 2 d = c c c c x = vt x = vt mirrors d = c apart time out and back (1 tick) = 2 clock records wristwatch time You see the light take a longer path but the speed is still c So the time t is longer. The moving clock ticks more slowly. time dilation t = t = with substitute distance x = vt 2 = t 2 (1 – v2/c2 ) gives 1 1 – v 2 /c 2 t= 1 – v 2 /c 2 Time dilation is a consequence of the constant speed of light In this case, the clock going past takes a longer time t for each tick than its own ‘wristwatch’ time τ. The relation between the two is t =γτ where 1 1 v2 / c2 The time dilation effect has been confirmed experimentally using muons (see Question 120S). Examples of time dilation in practice are given on page 82 of the student text. Page | 20 The following activities are designed to support the work on time dilation. Activity 110S Software-based ‘The light clock’ When you run the clock you should be able to understand how it works so that you can see that clocks which move past you must tick slower than any you carry with you showing your own 'wrist watch time'. Activity 60S Software-based ‘Investigating the time dilation factor γ’ Here you study the way the time dilation factor varies with the ratio v / c. You can look for ranges of values in which it is undetectable, and ranges of values for which it is large. Page | 21 Ch 12.3 Was there a ‘Big Bang’? Red shifts of distant galaxies give evidence of the expansion of the Universe. A red shift z corresponds to an expansion in scale of RNOW 1 z RTHEN Current estimates of the expansion time-scale of the Universe put it at about 14 ± 2 Gy. Evidence that the Universe has evolved from an initial uniform, hot dense state comes from the existence of the cosmic microwave background. There are still fundamental problems in explaining the major features of the Universe. An expanding Universe We present just two key pieces of evidence for the expansion and origin of the Universe. The two key ideas are the cosmological red-shift and the cosmic microwave background, as evidence of the expansion of the Universe from a ‘hot big bang’ origin. No reference is made to the relative proportions of the elements in the early Universe. It is important to appreciate that at large red shifts, we are not looking at ‘recession velocities’ in any simple sense. The space of the whole Universe is expanding, stretching wavelengths with it. Thus Doppler shifts, which do indicate relative velocities for nearer objects, now indicate an expansion of space. Hubble established the relation between red-shift and distance but the value for the Hubble constant has made rather dramatic changes in its value over time. A larger value of H0 would mean a more rapidly expanding Universe. v = H0r (v = recession velocity in kms-1 and r = distance in Mpc where 1 Mpc = 3.09 1019 m) Notes: Page | 22 Some experimental evidence of the relation between the red-shift and the distance to five galaxies is shown below (see p 84). Display Material 180O OHT 'Red shifts of galactic spectra’ The photographs are of the brightest galaxies in successively more distant clusters, together with observed red shifts in the light from these galaxies. The two dark absorption lines are due to calcium – they are called the H and K lines. Page | 23 There are real and fundamental difficulties involved in the measurement of distances on this scale. As a consequence the ‘accepted’ value of the Hubble constant has changed many times. Display Material 140O OHT 'How the accepted value of the Hubble constant has changed' How estimates of the Hubble time have changed + Hubble 1929 600 + + + 2 + 400 + + 200 + age of Earth 4.6 Gyr + + 100 + + ++ over 300 m easurements in disputed range 50–100 5 + + 10 20 de Vaucouleurs current best value 13.7 G yr 0 1920 1940 1960 1980 2000 Sandage year Cosmic convergence – the graph shows the evolution of the Hubble constant measurements from the 1920s until the present day. Over 300 measurements of the Hubble constant, H0, have been published since 1975. Recent results are converging towards a value of 65 km s-1 Mpc-1. Page | 24 Display Material 160O OHT 'Hubble's law and the age of the Universe' Hubble’s law and the age of the Universe Hubble found that the further away a galaxy is, the larger its redshift. He interpreted this to mean that distant galaxies are receding from us. For a galaxy a distance d from us, Hubble wrote v=Hd where v is the speed of a galaxy away from us and H is a constant called Hubble’s constant. Run the Universe backwards in time... ...distant galaxies are further away but are moving faster... ...in the past galaxies must have been closer together... ...even further back, all the matter and space in the Universe was concentrated at a single point. A galaxy distance d from us takes a time t = d/v to reach us in a reversed Universe. From Hubble’s law: d d 1 t=v= = Hd H This time is independent of d and v and tells us how long ago the Universe was a single point - this is the age of the Universe. Strictly, in a reversed Universe, the galaxies accelerate as they fall together so that the ‘Hubble time’, 1/H, gives an upper limit for the age of the Universe. Page | 25 Display Material 200O OHT 'The age of the Universe' The ‘age of the Universe’ + + + + + + + + distance to galaxy speed of recession is directly proportional to distance v = H0 r 1 = Hubble time H0 H0 is the Hubble constant units of H0 are speed 1 = time distance If speed v were constant, then units of 1 = time H0 1 r = = time since galaxies were close together H0 v If we assume that the speed v was constant, the Hubble time would represent the time since galaxies were close together. This time represents the age of the Universe. Exercise (a) Show that a Hubble constant of 65 km s-1 Mpc-1 leads to an estimate of 15 Gyr for the age of the Universe. [Hint: take great care with the units] ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… Page | 26 (b) Suggest why the Universe must be younger than this time. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… Notes: Page | 27 Display Material 150O OHT 'The history of the Universe' These timelines in two different forms show the history of the Universe on a logarithmic scale, illustrating important events. Times since the Big Bang / s 101 107 1013 108 1014 103 109 1015 104 1010 1016 105 1011 1017 10 2 He nuclei formed. Proportions of matter in the Universe fixed Neutral atoms formed. Photons travel freely through the Universe Galaxies formed Solar system formed Life arises on Earth Extinction of the dinosaurs The present 106 1012 1018 Page | 28 A spiral timeline showing the history of the Universe on a logarithmic scale. The history of the Universe M oving inwards along the tim e spiral, eac h point is one tenth of the age of the previous one, m easured in s econds . T he c urve spirals endles sly inw ards to the Big Bang. Neutral a tom s form ed. Photons trav el freely through the U nivers e 10 14 10 6 10 15 10 13 10 7 10 5 10 16 1 0 8 10 4 10 12 10 2 10 9 10 3 Helium nuc lei form ed Galaxies formed C urrent theories do not go nearer the Big Bang than 10 –36 s 1 0 17 Solar s ys te m form ed 10 11 10 10 10 18 N ow Extinction of the dinosaurs Page | 29 Display Material 170O OHT 'Relativity and the expanding Universe' The expanding Universe according to general relativity According to general relativity, the Big Bang was not an explosion of matter into empty space. Both space and matter came into existence together about 14 billion years ago. Imagine a balloon with dots drawn on it to represent galaxies, and slowly blow it up. As the balloon grows, the space between the galaxies grows. The galaxies do not move within the surface. In the real Universe - unlike in the balloon model - the galaxies themselves do not grow: their gravity holds them together. General relativity pictures the expansion of spacetime as if it were an expanding balloon. This animation helps to bring alive the idea of an expansion of space itself. Activity 130S Software Based 'The cosmological red shift' Using File 70L Launchable File 'The space expands' which is launched from the CD ROM \components\Flash\120079f1.exe Page | 30 The cos mological red-shift... Think of an electromagnetic wave drawn on the balloon, travelling from one galaxy to another. A light wave travels from galaxy 1 to galaxy 2. T he galaxies are a distance de mitte d apart when the wave is emitted. The Universe expands. em itted d em itted Space is stretched and the wave with it. rec eiv ed d rece ived When the light is received, the galaxies are a distance d receiv ed apart. Wavelengths are red shifted because spacetime stretches as the light travels through it. The expansion of space is related to the cosmological red shift. Page | 31 Cosmological red-shift Light travels from one galaxy to another, as the Universe expands light em itted Space stretching Wavelength stretching em itted em itted R emitted observed R emitted R observed R observed = R emitted observed em itted The Universe expands...the photons travel observed light observed R observed R emitted R observed R emitted R observed R observed R emitted = em itted + em itted =1+ emitted observed = emitted + =Z emitted =1+z Question Radio galaxy 3C324 has a red shift z = 1.12. Calculate by what percentage the universe has expanded since the light from it was emitted. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… Evolution from hot to cool The discovery of the cosmic microwave background radiation is outlined on pp 87-89. This was interpreted as the expanded (red shifted) radiation from the time around 300 000 years after the ‘big bang’ when ionised atoms recombined into atoms, and photons no longer interacted strongly with them. This gives evidence of an evolving Universe; one that was not the same in the past as it is now. This appeared as a bit of radio interference that wouldn’t go away, yet it swung opinion towards an idea that several had found absurd, even self-contradictory – that space-time had a beginning. Page | 32 Display Material 210O OHT 'The cosmic microwave background radiation' The cosmic microwave background radiation In the beginning... ...there was the Big Bang... ... the Universe is filled with a plasma of elementary particles, all exchanging energy with photons of electromagnetic radiation. + + + + + + ...300 000 years after the Big Bang... Temperature: 3000 K Typical wavelength of radiation: 1 m As the temperature falls, atoms form as electrons are held in orbit around nuclei of protons and neutrons. The Universe becomes transparent to photons which no longer interact so easily with atoms and so travel unaffected through the Universe. + + + + + The decoupling of the radiation – the Universe becomes transparent to electromagnetic radiation. Page | 33 The cosm ic microwave background radiation Interstellar space is filled with a photon ‘gas’ (and some atoms). The temperature of this gas is proportional to the energy of the photons. The energy of a photon is proportional to its frequency. Therefore the temperature of the photon gas is proportional to the frequency of the radiation. ...13 billion years after the Big Bang. Temperature: 2.7 K Typical wavelength of radiation: 1 mm The Universe expands, stretching the wavelength of the photons. The greater the wavelength, the lower the frequency. The temperature of the photon gas falls. The expansion of the Universe stretches the wavelength of the radiation, decreasing its frequency and therefore reducing the energy density and lowering the temperature. Question (a) Use the information given above to argue that the wavelength of the radiation is inversely proportional to the temperature. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… Page | 34 (b) Use the inverse relationship between wavelength and temperature, and information supplied in Display Material 120O to show that infra-red radiation of wavelength 1 μm formed 300 000 years after the Big Bang will now be in the microwave part of the electromagnetic spectrum. ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… The expansion of the Universe T he cosmic microwav e background radiation no w: cluster of galaxies 0 R0 typical distance that increases with the expansion of the Universe, R0 typic al wavelength, 0 then: Rt t typic al wavelength, t typical distance between clusters, R t 0 > t R0 > Rt so put 0 = t + 0 R0 Rt R0 Rt t = t + t = = 1+ = t 1+ z This is the same effect as is seen in the r ed-shift of distant galaxies. The expansion of spacetime and the stretching of wavelengths compared. Page | 35 Display Material 220S COBE' Computer Screen 'The cosmic microwave background – This image is a COBE image of the microwave background showing the temperature distribution. The temperature distribution of the cosmic microwave background radiation is over the whole celestial sphere, corrected for the effect of the Earth’s motion. See page 89 for a colour version. As mentioned in the introduction to this section, we have now presented the two key pieces of evidence for the expansion and origin of the Universe: cosmological red-shift and the cosmic microwave background. You may have time to do Activity 140P Presentation ‘The universe’ which will involve a bit of independent research. The Classroom Video ‘The Big Bang’ provides a good summary for much of the work of this chapter. Page | 36 Questions and activities additional to those listed in the Student Notes Section Essential Optional 12.1 Read A2 text pp 65-73 Qu 1-6 A2 text p 74 Question 20W Warm-up exercise 'Using time to measure distance' Question 30W Warm-up exercise 'Units for distance measurement' Question 52S Short Answer 'Doppler detection' Question 55S Short Answer 'Doppler shifts in astronomy' Question 60S Short Answer 'Binary stars' Question 20S Short Answer 'Measuring distances within the solar system and beyond' Question 40S Short Answer 'Comparing intensities for lamps' Question 45S Short Answer 'Jupiter and Saturn close together in the sky' Question 46S Short Answer 'Brighter stars aren’t always nearer' Question 50S Short Answer 'Trip times tell distances' Question 30C Comprehension 'Apparent star brightnesses and logarithmic scales' (hard) Question 70D Data Handling 'Using orbital data to calculate masses' Question 10X Explanation–Exposition 'Logarithmic scales' Reading 20T Text to Read 'The ladder of astronomical distances' Display Material 80S Computer screen ‘Radar images: volcano’ Display Material 100S Computer screen ‘Magic from trip times’ File 10S Speadsheet datatable ‘Magnitude and brightness’ File 20I farther’ Image ‘Looking for longer, seeing Activity 70E Experiment 'Investigating the measurement of distance using an ultrasonic sensor' Activity 80E Experiment 'Tap-tap range finding' Activity 60H Home Experiment 'Two-million year old light: Seeing the Andromeda nebula' The following case studies are relevant here: ‘Missing mass in the Universe’ ‘Profiles from space’ These can be found in ‘Case studies: advances in physics’ at the back of the A2 text. 12.2 Read A2 text pp 75-82 Qu 1-6 A2 text p 83 Question 100C Comprehension ‘The MichelsonMorley experiment’ Question 50C Comprehension ‘Thinking relatively’ Question 75S Short Answer 'Relativistic Doppler effect assuming time dilation' Question 85S Short Answer 'Light clocks and time dilation' Question 150S Short Answer 'Time dilation and length contraction for particles' Question 50W Warm-up exercise 'When does the speed of light matter' Question 60W Warm-up exercise 'The relativistic time-dilation factor γ' Question 70S Short Answer 'Practice with the relativistic Doppler shift equation' Question 80S Short Answer 'Practice with the relativistic time dilation equation' Display material 115O OHT ‘Two-way radar speed measurement 2’ Display material 135O OHT ‘Time dilation at v/c = 3/5’ Display material 137O OHT ‘Relativistic addition Page | 37 Question 120S Short Answer 'Time dilation for muons' 12.3 Read A2 text pp 84-89 Qu 1-6 A2 text p 90 Question 40W Warm-up exercise 'Cosmological expansion' Question 95S Short Answer 'Redshifts of quasars' Question 100S Short Answer 'cosmic microwave background radiation' of velocities’ Reading 40T Text to read ‘Einstein’s 1905 relativity paper’ Reading 100T Text to read ‘Why we believe in special relativity: experimental support for Einstein’s theory’ Reading 100L Book list ‘Books about relativity’ Display Material 230S Computer Screen 'Looking out for longer' Display Material 240S Computer Screen 'The Universe at different wavelengths' Reading 50T Text to Read 'The breakthrough of the year' Reading 60T Text to Read 'The sky is dark at night: A reason to think the Universe has evolved' Reading 70T Text to Read 'Life CV: Edwin Powell Hubble' Question 80D Data Handling 'Astronomical distances' Question 90D Data Handling 'Calculating the age of the Universe' Question 110C Comprehension 'Evidence for a hot early Universe’ Summary Qu 1-5 A2 text p 92 These notes draw almost exclusively on the resources to be found in Advancing Physics A2 Student’s Book and CD-ROM published by Institute of Physics Publishing in 2000 and 2008. They are intended to be used in conjunction with these resources and others not specified. John Mascall The King’s School, Ely, Cambs Page | 38