Chapter 3 Cosmology 3.1 The Doppler effet Learning objeties: Why does the waelength of waes from a moing soure depend on the speed of the soure? What is a Doppler shift? How an we measure the eloity of the two stars in a binary system? Doppler shifts The waelengths of the light waes from a star moing towards the Earth are shorter than they would be if the star was stationary. If the star had been moing away from the Earth, the waelengths of the light waes from it would be longer than if the star was stationary. This effet applies to all waes and is known as the Doppler effet. It is the reason why the pith of a siren on an approahing emergeny ehile rises as the ehile approahes then falls sharply as it passes by. The pith is higher as the soure approahes and lower as it retreats beause the soure moes a ertain distane eah time it emits eah yle of waes. Consider a soure of waes of frequeny f moing at speed. Figure 1 shows wae fronts representing suessie wae peaks emitted by the soure at time interals t = f 1. Figure 1 The Doppler effet
The distane between suessie wae peaks is the waelength of the waes. In time t, eah wae peak shown traels a distane t (= f ) before the next wae peak is emitted and the soure traels a distane t (= f ). Waes emitted in the opposite diretion to the motion of the soure ( behind the soure) are spaed out. An obserer in the path of these waes would therefore detet waes of longer waelength and therefore lower frequeny. For light, this shift to longer waelengths is referred to as a red shift as it auses the lines of a line spetrum to shift towards the red end of the isible spetrum. Waes emitted in the same diretion as the motion of the soure are bunhed together ahead of the soure. An obserer in the path of these waes would therefore detet waes of shorter waelength and therefore higher frequeny. For light, this shift to longer waelengths is referred to as a blue shift as it auses the lines of a line spetrum to shift towards the blue end of the isible spetrum. It an be shown that for a soure moing at speed relatie to an obserer, towards the obserer: the hange of frequeny f = the hange of waelength = away from the obserer: the hange of frequeny f = the hange of waelength = f f f The Doppler shift, z, in frequeny (or waelength) is the frational hange (or ). f Mathematially, red shifts and blue shifts are frational hanges in frequeny or waelength. Table 1 summarises the frational hange, z, in frequeny and waelength. Doppler shift, z in frequeny f f Soure moes towards obserer + Soure moes away from obserer in waelength + Table 1 Summary of Doppler shifts The frequeny f is the frequeny of the light emitted by the soure whih is the same as the frequeny emitted by an
idential soure in the laboratory. The hange of frequeny f is the differene between this frequeny and the obsered frequeny (the frequeny of the light from the soure as measured by an obserer). Notes 1 The formulas aboe and in Table 1 an only be applied to eletromagneti waes at soure speeds muh less than the speed of light. 2 A star or galaxy may be moing through spae with perpendiular eloity omponents parallel and at right angles to the line from the Earth to the star. The first omponent is the star s radial speed and the seond omponent its tangential speed. Throughout this topi, speed refers to its radial speed (i.e. the omponent of the star s eloity parallel to the line between the star and the Earth). Astronomial eloities The line spetrum of light from a star or galaxy is shifted to longer waelengths if the star or galaxy is moing away from us and to shorter waelengths if it is moing towards us. By measuring the shift in waelength of a line of the star s line spetrum, the speed of the star or galaxy relatie to Earth an be found. If the star is part of a binary system, its orbital speed an also be found as explained later. In pratie, the line spetrum of the star or galaxy is ompared with the pattern of the prominent lines in the spetrum aording to the star s spetral lass. The hange of waelength of one or more prominent lines of known waelength in the spetrum is then measured and the Doppler shift, z (= ) is then alulated. For an indiidual star or galaxy, the speed of the star or galaxy relatie to a line between Earth and the star is then alulated from z =. The star or galaxy is moing: towards the Earth if the waelength is shortened due to the star or galaxy s relatie motion away from the Earth if the waelength is lengthened due to the star or galaxy s relatie motion. For binary stars in orbit about eah other in the same plane as the line from the Earth to the stars, the waelength of eah spetral line of eah star hanges periodially between: a minimum alue of when the star is moing towards the Earth a maximum alue of + when the star is moing away from the Earth. Worked example A spetral line of a star is found to be displaed from its laboratory alue of 434 nm by +0.087 nm. State whether the star is moing towards or away from the Earth and alulate its speed relatie to the Earth. = 3.0 10 8 m s 1 Solution The star is moing away from the Earth beause the waelength of its light is inreased. Rearranging 3.010 0.087 10 gies 9 43410 8 9 6.010 4 ms 1
If the two stars annot be resoled, they are referred to as a spetrosopi binary. Eah spetral line splits into two after the stars ross the line of sight then merge into a single line as the two stars moe towards the line of sight. Figure 2 shows the idea. Figure 2 A spetrosopi binary Note If the stars are of different masses, they will moe with the same period but at different speeds and orbital radii. The hange of waelength will be greater for the faster star (less massie star) than for the other star. Worked example = 3.0 10 8 m s 1 A spetral line of a ertain spetrosopi binary merges one eery 1.5 years and splits to a maximum displaement of 0.042 nm and 0.024 nm from their laboratory waelength of 486 nm. Calulate: a the orbital speed of eah star b the radius of orbit of the larger orbit. Solution a For the slower star, = 0.024 nm Rearranging 3.010 0.024 10 gies 9 48610 For the faster star, = 0.042 nm 8 9 1.510 4 ms 1 Rearranging 3.010 0.042 10 gies 9 48610 8 9 2.610 b The orbital speed of the faster star, = 2r/T where r is its radius of orbit and T is the time period. 4 ms 1
4 1 T 2.6 10 ms 1.5 365.25 246060s Therefore, the radius of its orbit Hene r = 2.0 10 11 m r 2π 2π Summary questions = 3.0 10 8 m s 1 1 Explain why the waelengths of the light waes from a star moing away from the Earth are longer than they would be if the star was stationary relatie to the Earth. 2 A spetral line of a star is found to be displaed from its laboratory alue of 656 nm by 0.035 nm. State whether the star is moing towards or away from the Earth and alulate its speed relatie to the Earth. 3 The spetral lines of a star in a binary system ary in waelength. a Explain why this ariation is: i periodi ii oer a narrow well-defined range of waelengths. b i State what measurements an be made by obsering the ariation in waelength of a spetral line from suh a star. ii Explain how the measurements an be used to find the radius of orbit of the star. 4 A spetral line of a ertain star in a binary system hanges from its laboratory waelength of 618 nm by ±0.082 nm with a time period of 2.5 years. Calulate: a the orbital speed of the star b its radius of orbit.
3.2 Hubble s law and beyond Learning objeties: What do we mean by the term red shift? Why do we think the Unierse is expanding? What eidene led to the aeptane of the Big Bang theory What is dark energy? Galaxies The Andromeda galaxy is the nearest large galaxy to the Milky Way. Andromeda an just be seen by the unaided eye on a lear night. By taking photographs of Andromeda using a large telesope, Edwin Hubble was able to identify Cepheid ariable stars in Andromeda. These stars ary in brightness with a period of the order of days and are named after the first one to be disoered, -Cephei, the fourth brightest star in the onstellation Cepheus. Their signifiane is that the period depends on the absolute magnitude. Hubble measured the periods of the Cepheid ariables in Andromeda that he had identified. He then used data obtained on Cepheid ariables of known absolute magnitudes to find the absolute magnitude and hene the distane to eah Cepheid ariable in Andromeda. He found that Andromeda is about 900 kiloparse away, far beyond the Milky Way galaxy whih was known to be about 50 kiloparse in diameter. His result settled the issue of whether or not Andromeda is inside or outside the Milky Way galaxy. Astronomers realised that many spiral nebula they had obsered like Andromeda must also be galaxies. The Unierse onsists of galaxies, eah ontaining millions of millions of stars, separated by ast empty spaes. Hubble and other astronomers studied the light spetra of many galaxies and were able to identify prominent spetral lines as in the spetra of indiidual stars but red-shifted to longer waelengths. Hubble studied galaxies whih were lose enough to be resoled into indiidual stars. For eah galaxy, he measured: its red shift and then alulated its speed of reession (the speed at whih it was moing away) its distane from Earth by obsering the period of indiidual Cepheid ariables in the galaxy. His results showed that galaxies are reeding from us, eah moing at speed whih is diretly proportional to the distane, d. This disoery, referred to as Hubble s law, is usually expressed as the following equation: = Hd where H, the onstant of proportionality, is referred to as the Hubble onstant. For distanes in megaparse (Mp) and eloities in km s 1, the aepted alue of H is 65 km s 1 Mp 1. In other words, the speed of reession of a galaxy at a distane of: 1 Mp is 65 km s 1 Mp 1 10 Mp is 650 km s 1 Mp 1 100 Mp is 6500 km s 1 Mp 1
Figure 1 shows that the pattern of typial measurements of the speed of reession and distane d plotted on a graph is a straight line through the origin. (This example is from atual data published in 2002. Notie the error bars on the distane estimates.) The slope of the graph is equal to the Hubble onstant H. Figure 1 Speed of reession against distane for galaxies Note The galaxies loal to the Milky Way galaxy suh as Andromeda do not fit Hubble s law beause their graitational interations hae affeted their diretion of motion. Andromeda is known to be on ourse to ollide with the Milky Way galaxy billions of years in the future. Worked example = 3.0 10 8 m s 1, H = 65 km s 1 Mp 1 The waelength of a spetral line in the spetrum of light from a distant galaxy was measured at 398.6 nm. The same line measured in the laboratory has a waelength of 393.3 nm. Calulate: a the speed of reession of the galaxy b the distane to the galaxy. Solution a = 398.6 393.3 = 5.3 nm Rearranging 3.010 5.310 gies 9 39310 b Conerting to km s 1 gies = 4.0 10 3 km s 1 8 9 4.010 6 ms 1 Rearranging = Hd gies d H 1 4.010 kms 1 65kms Mp 3 1 62Mp
The Big Bang theory Hubble s law tells us that the distant galaxies are reeding from us. The onlusion we must draw from this disoery is that the galaxies are all moing away from eah other and the Unierse must therefore be expanding. At first, some astronomers thought this expansion is beause the Unierse was reated in a massie primordial explosion and has been expanding eer sine. This theory was referred to by its opponents as the Big Bang theory. With no eidene for a primordial explosion other than an explanation of Hubble s law, many astronomers supported an alternatie theory that the Unierse is unhanging, the same now as it eer was. This theory, known as the Steady State theory, explained the expansion of the Unierse by supposing matter entering the Unierse at white holes pushes the galaxies apart as it enters. The Big Bang theory was aepted in 1965 when radio astronomers disoered mirowae radiation from all diretions in spae. Steady state theory ould not explain the existene of this mirowae radiation but the Big Bang theory ould. Estimating the age of the Unierse The speed of light in free spae,, is 300 000 km s 1. No material objet an trael as fast as light. Therefore, een though the speed,, of a galaxy inreases with its distane d, no galaxy an trael as fast as light. The Hubble onstant tells us that the speed of a galaxy inreases by 65 km s 1 for eery extra million parses of distane or 3.26 million light years. Therefore, a galaxy traelling almost at the 300 000 speed of light would be almost at a distane of 3.26 million light years. 65 To reah this distane, light would need to hae traelled for 15 000 million years. Thus the Unierse annot be older than 15 000 million years. Note In mathematial terms, the speed of a galaxy < Therefore, using the equation for Hubble s law gies Hd < or d < H The distane H represents the maximum expansion of the Unierse and light ould not hae traelled further than this distane sine the Unierse began. The age of the Unierse, T, is therefore gien by equating the distane traelled by light in time T (= T) to the expansion distane H. Hene T = H gies 1 T H Substituting H = 65 km s 1 Mp 1 = 2.1 10 18 s 1 (as 1 Mp = 3.1 10 22 m) therefore gies 1 1 T = 4.7 10 17 s = 15 000 million years. H 18 1 2.110 s
Eidene for the Big Bang theory The spetrum of mirowae radiation The spetrum of mirowae radiation from spae mathed the theoretial spetrum of thermal radiation from an objet at a temperature of 2.7 K. Beause the radiation was deteted from all diretions in spae with little ariation in intensity, it was realised it must be uniersal or osmi in origin. This bakground osmi mirowae radiation is explained readily by the Big Bang theory as radiation that was reated in the Big Bang has been traelling through the Unierse eer sine the Unierse beame transparent. As the Unierse expanded after the Big Bang, its mean temperature has dereased and is now about 2.7 K. The expansion of the Unierse has gradually inreased the bakground osmi mirowae radiation to its present range of waelengths. Relatie abundane of hydrogen and helium Stars and galaxies ontain about three times as muh hydrogen by mass as helium. In omparison, other elements are present in negligible proportion. This 3 : 1 ratio of hydrogen to helium by mass means that for eery helium nuleus (of mass 4 u approximately) there are 12 hydrogen nulei (of mass 12 u in total). Thus there are 14 protons for eery 2 neutrons (proton : neutron ratio of 7 : 1) is ratio is beause the rest energy of a neutron is slightly greater than that of the proton. As a result, when the Unierse ooled suffiiently to allow quarks in threes to form baryons, protons formed from the quarks more readily than neutrons. Preise alulations using the exat differene in the rest energies of the neutron and the proton yield a 7 : 1 ratio of protons to neutrons. Figure 2 Formation of hydrogen and helium nulei
Dark energy Astronomers in 1998 studying type Ia supernoa were astounded when they disoered ery distant supernoae muh further away than expeted. To reah suh distanes, they must hae been aelerating. The astronomers onluded that the expansion of the Unierse is aelerating and has been for about the past 5000 million years. Before this disoery, most astronomers expeted that the Unierse was deelerating as ery distant objets would be slowed down by the fore of graity from other galaxies. Many more obserations sine then hae onfirmed the Unierse is aelerating. Sientists think that no known fore ould ause an aeleration of the expansion of the Unierse and that a hitherto-unknown type of fore must be releasing hidden energy referred to as dark energy. Eidene for aelerated expansion of the Unierse is based on differing distane measurements to type Ia supernoa by two different methods: 1 The red shift method: measurement of the red shift of eah of these distant type Ia supernoa and use of Hubble s law gies the distane to eah one. 2 The luminosity method: Type Ia supernoa at peak intensity are known to be 10 9 times more luminous that the Sun, orresponding to an absolute magnitude of about 18. The distane to suh a supernoa an be alulated from its absolute magnitude M and its apparent magnitude m using the formula m M = 5 log d. 10 The two methods gie results that are different and indiate that the distant type Ia supernoa are dimmer and therefore further away than their red shift indiates. The nature of dark energy is unlear. It is thought to be a form of bakground energy present throughout spae and time. It is more prominent than graity at ery large distanes beause graity beomes weaker and weaker with inreased distane whereas the fore assoiated with dark energy is thought to be onstant. Current theories suggest it makes up about 70% of the total energy of the Unierse. The searh for further eidene of dark energy will ontinue with obserations using larger telesopes and more sensitie mirowae detetors on satellites. Summary questions = 3.0 10 8 m s 1, H = 65 km s 1 Mp 1 1 a State Hubble s law. b Explain why Hubble s law leads to the onlusion that the Unierse is expanding. 2 The waelength of a spetral line in the spetrum of light from a distant galaxy was measured at 597.2 nm. The same line measured in the laboratory has a waelength of 589.6 nm. Calulate: a the speed of reession of the galaxy b the distane to the galaxy. 3 State two piees of experimental eidene other than Hubble s law that led to the aeptane of the Big Bang theory of the Unierse. 4 A ertain type Ia supernoa has an apparent magnitude of +24. a Calulate the distane to the supernoa. (Assume the absolute magnitude of any type Ia supernoa is 18.) b Outline why measurements on type Ia supernoae hae led to the onlusion that the expansion of the Unierse is aelerating.
3.3 Quasars Learning objeties: How were quasars disoered? What are the harateristi properties of a quasar? Why are there no nearby quasars? The first quasar The first quasar was announed two years after a preiously disoered astronomial radio soure, 3C 273, was identified as a dim star in 1962 using an optial telesope. The star presented a puzzle beause its radio emissions were stronger than expeted from an ordinary star and its isible spetrum ontained strong lines that ould not be explained. Astronomers in California realised the strong lines were due to a ery large red shift of 0.15, orresponding to a light soure with speed of reession of 0.15 (i.e. 15% of the speed of light) at a distane of oer 2000 million light years away. Based on this distane, alulations showed that 3C 273 is 1000 times more luminous that the Milky Way galaxy yet ariations in its brightness indiated it is muh smaller than the Milky Way galaxy. Its ariations on a time sale of the order of years or less tell us that its diameter annot be muh more than a few light years. Astronomers onluded that 3C 273 is more like a star than a galaxy in terms of its size yet its light output is on a galati sale or een greater. The objet was referred to as a quasi-stellar objet or quasar. Many more quasars hae been disoered moing away at speeds up to 0.85 or more at distanes between 5000 and 10 000 light years away. The absene of quasars loser than about 5000 million light years indiates a quasar age that ommened 2000 to 3000 million years after the Big Bang and lasted about 5000 million years. Notes 1 To alulate the red shift of a quasar, the hange of waelength of one of its spetral lines of known waelength is measured and then used to alulate the red shift z. 2 To alulate the speed of reession, the equation = z may be used only if <<. Otherwise, a relatiisti equation relating and z must be used. Knowledge of this relatiisti equation is not required in this option speifiation. Quasars generally hae red shifts between 1 and 5 orresponding to speeds from 0.6 to about 0.95 whih are not insignifiant ompared with the speed of light,. Quasar properties Quasars are among the oldest and most distant objets in the Unierse. A quasar is haraterised by: its ery powerful light output, muh greater than the light output of a star its relatiely small size, not muh larger than a star a large red shift indiating its distane is between 5000 and 10 000 light years away. Many quasars are not like 3C 273 in that they do not produe strong radio emissions.
What are quasars? Detailed optial and radio images of quasars indiate fast-moing louds of gases and jets of matter being ejeted. Quasars are found in or near galaxies whih are often distorted, sometimes with lobes either side. Suh atie galaxies are thought to hae a supermassie blak hole at their entres. As disussed in Topi 2.4, suh a blak hole ould hae a mass of more than 1000 million solar masses. With many stars near it, matter would be pulled in and would beome ery hot due to ompression as it nears the eent horizon. Oerheating would result in louds of hot glowing gas being thrown bak into spae. A spinning supermassie blak hole would emit jets of hot matter in opposite diretions along its axis of rotation. Many astronomers think that a quasar is a supermassie blak hole at the entre of a galaxy. When we obsere a quasar, we are looking bak in time at a supermassie blak hole in ation. The ation eases when there are no nearby stars for the blak hole to onsume. Fortunately, the Milky Way galaxy and Andromeda and other galaxies lose to us are relatiely inatie beause eah galaxy no longer has many stars left near the supermassie blak hole at its entre. Summary questions = 3.0 10 8 m s 1, H = 65 km s 1 Mp 1 1 State three harateristis of a quasar. 2 Light from a ertain quasar was found to ontain a spetral line of waelength 540 nm that had been red-shifted from a normal waelength of 486 nm. a Show that the red shift of this quasar is 0.11. b Calulate the speed of reession of this quasar, assuming its speed is muh less than the speed of light. Ignore relatiisti effets. 3 a What features of the light from a quasar indiates a quasar is muh more luminous than a star? b What feature of the light from a quasar indiates a quasar is muh smaller in size than a galaxy? 4 Outline why astronomers think ertain galaxies hae a supermassie blak hole at their entres.
Answers 1.1 1 a i The top ray should refrat at the lens and pass through F; the bottom ray should refrat at the lens and then beome parallel to the prinipal axis; the image should be formed at 1.4(3)f on the right-hand side of the lens. ii real, inerted, diminished b ii irtual, magnified, upright 2 a and = + 0.240 m b i real 3 a and = 0.300 m b i irtual ii inerted ii upright 4 a i = 0.600 m, image height = 40 mm ii = +1.000 m, image height = 40 mm 1.2 b i 0.450 mm, irtual and upright 2 b i 7.5 ii 1.13 (1.125 to 4 s.f.) ii 1.250 m, real and inerted 4 a i 640 mm ii 680 mm b i 0.3 ii 1.4(3) 10 4 m 1.3 4 a 100 b 40 1.4 3 a 0.06(3) m 4 b 80 m 1.5 2 a i 2 10 5 degree ii 3 10 6 degree 4 a 4, 1, 2/3, 5 2.1 2 b ii +11.63 3 b +5.9 4 b 23.2 2.2 2 b 4700 K 4 b 2.0 10 9 m 2.3
4 a Z, Y, X b X giant; Y main sequene; Z white dwarf X/Z = 6300, Y/Z = 100 2.4 4 b i 3.0 km ii 1.8 10 19 kg m 3 1.8 10 5 3.1 2 1.6 10 4 m s 1 4 a 4.0 10 4 m s 1 b 5.0 10 11 m 3.2 2 a 3.9 10 6 m s 1 b 59 Mp 4 a 2500 Mp Image aknowledgements Setion 1.4 figure 4: NASA, ESA, and the Hubble Heritage Team (STSI/AURA) Setion 2.4 figure 1: NASA, ESA, J. Hester and A. Loll (Arizona State Uniersity) Setion 2.5 figure 1: ESO; figure 3: CSIRO; figure 4: ESA Setion 3.2 figure 1: PNAS