Physics resource created during Science Degree
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The Age of the Universe Physics 210 – Laboratory 3 Richard Laugesen, 30/9/99 Richard Laugesen Page 2 Introduction The universe is believed to have started from a point of space-time thought to be similar to a singularity. It exploded into existence in a big bang, and since then all matter has been expanding. From our frame of reference the rest of the universe appears to be moving away from us. Either we are at the centre of an expanding universe or everything everywhere is expanding away from its neighbours. If the big bang theory is believed to be true (which it commonly is) then the second argument is the correct one. A quick thought experiment. Suppose a universe was 1 metre in diameter, an observer on one side of the proto universe would easily see the other side with no noticeable delay. Now suppose that this universe has expanded to a diameter of 10 billion meters, a distance impossible to comprehend, the observer would now notice a significant delay in her observation of the other side of the universe. This effect is due to the finite speed of light. From the edge of our observable universe it takes light approximately 15 billion years to reach earth, so anything we observe at that distance happened 15 billion years ago, or in other words close to the beginning of the universe. It took cosmologists half a century to grasp the full cosmological importance of a finite speed of light. It divides the universe into two parts: visible and invisible. At any time there is a spherical “horizon” around us, defined by the distance light has been able to travel since the universe began. As time passes, this horizon expands. Today, it is about 15 billion light years away. As the universe exploded from the big bang it was accelerated. Matter further from the earth has had more time to accelerate and as such has a greater velocity, also true for galaxies, which were observed in this experiment. A greater velocity results in a larger redshift of any electromagnetic waves emitted from an object, such that all the wavelengths would be shifted to a longer wavelength (or towards the red end of the spectrum). This is due to the Doppler effect. From the redshift of absorption and emission lines in a galaxies’ spectrum it is possible to deduce the velocity that it is receding. In this experiment the H and K Calcium lines were used, which normally appear at 3968.847 Å and 3933.67 Å respectively. The Hubble constant can then be derived from the linear trend of the Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 3 velocity of a number of galaxies. Once an approximation of the Hubble constant is known an estimate on the age of the universe is just another small step. Definitions Absolute Magnitude The magnitude a star would have if it were precisely 10 parsecs away from the sun. Angstrom Å The unit normally used to measure wavelengths of visible and ultraviolet light. One angstrom is 10-8 Cm. Apparent Magnitude The observed magnitude of a star or other object as seen from earth. Big Bang A term referring to any theory of cosmology in which the universe began at a single point, and has been expanding from that state since. Doppler Shift The observed shift in wavelength (and frequency) of a wave due to the relative motion between the source of the wave and the observer. Hubble Constant The numerical factor which describes the rate of expansion of the universe. Light year A unit of distance. The distance light has travelled in one earth year. Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 4 Neutron Star A very compact, dense remnant whose interior consists entirely of neutrons, which is supported against collapse by degenerate neutron gas pressure. Parsec The distance to a star if it has a parallax of 1 arcsecond. 3.03 x 1013 km. A mega parsec is 1 x 106 parsec Photon A particle of light having wave properties but also acting as a discrete unit. Pulsar An object that emits brief periodic pulses of radio energy, possibly a neutron star spinning very rapidly. Redshift The shifting of spectral lines toward the red end of the spectrum due to the relative motion of the source away from the observer. Relativistic When Einstein’s theory of relativity is used rather than classical theories. eg if an object is moving >1% speed of light. Singularity Spectra A point in space-time of infinite density. An arrangement of electromagnetic radiation according to wavelength. Signal-to-noise Is an indication of the amount of noise relative to the signal you want to measure. A value higher than 10 gives a fairly precise trace of the spectra when measuring stellar spectra. Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 5 Quasar A starlike object that has a very large redshift, is thought to be very distant, probably a galaxy emitting much more energy that is normal. Wavelength The distance between wavecrests in any type of Wave. Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 6 Results To calculate the distance to a galaxy the following equation was used m−M +5 5 Log D = Where m is the apparent magnitude, which is measured. M is the absolute magnitude, which is assumed. D is the distance. The wavelengths of the K and H lines of calcium were measured from the spectra of each galaxy. Spectra were obtained by collecting enough photons from the telescope to get a signal/noise ratio of 10 or greater on the spectrometer. These measured results were compared with the wavelengths of K and H lines obtained in the laboratory (with no Doppler shift) to calculate the velocity that the galaxy is receding using the following equation. ∆λ λ v=c where ∆λ = λmeasured − λ Where v is the velocity of the galaxy, c is the speed of light, λ is the wavelength with no Doppler shift and Δλ is the difference between the measured wavelength and the laboratory wavelength. The relativistic velocity was calculated using the following equation. ( z + 1) − 1 v = c ( z + 1) + 1 ∆λ λ where z= Where the variables are the same as above. The relativistic velocity uses Einstein’s Theory of General Relativity rather than Classical Theory to obtain a more accurate approximation when the velocity of the galaxy is greater than 1% of the speed of light. 1% the speed of light is 2998 km/s which all the measured galaxies at moving faster than. Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 7 Distance & Magnitude Measurements of Galaxy Galaxy Coma1 uma1-1 CrBor1 Boot1 uma2-1 Photon Count 1245978 304413 111526 92630 418874 Abs Mag -22.00 -22.00 -22.00 -22.00 -22.00 App Mag 12.80 14.50 15.40 16.50 16.60 Distance (Mpc) 91.20 199.53 302.00 501.19 524.81 Measurements of H & K Calcium Lines in Galaxy Galaxy Coma1 uma1-1 CrBor1 Boot1 uma2-1 λk measured 4022.00 4136.00 4220.00 4450.00 4474.00 λh measured 4058.00 4172.00 4256.00 4490.00 4514.00 Δλk -88.33 -202.33 -286.33 -516.33 -540.33 Δλh -89.15 -203.15 -287.15 -521.15 -545.15 Receding Velocity of Galaxy Galaxy Coma1 uma1-1 CrBor1 Boot1 uma2-1 Velocity of K (km/s) 6731.80 15419.95 21821.75 39350.49 41179.58 Velocity of H (km/s) 6734.30 15345.45 21690.51 39366.03 41178.90 Avg Velocity (km/s) 6733.05 15382.70 21756.13 39358.26 41179.24 Relativistic Receding Velocity of Galaxy Galaxy Coma1 uma1-1 CrBor1 Boot1 uma2-1 Velocity of K (km/s) 6807.36 15815.97 22613.69 41907.69 43977.20 Velocity of H (km/s) 6871.49 15881.91 22680.96 42322.74 44394.34 Avg Velocity (km/s) 6839.43 15848.94 22647.32 42115.21 44185.77 Physical Constants Speed Light λk λh 299792458 3933.67 3968.847 m/s Å Å Average Values of H Average Relativistic 77.47 82.47 km/s/mpc km/s/mpc *Determined from graph *Determined from graph * The velocity tables are absolute value. They are an indication of the galaxy receding at the given velocity not approaching, as a positive value could be interpreted as. Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 8 Receding Velocity of Galaxy Vs Distance 55000 50000 y =82.473x y =77.47x 45000 40000 Receding Velocity (km/s) 35000 30000 25000 20000 15000 Average Velocity Relativistic Velocity 10000 5000 0 0 100 200 300 400 500 600 Distance (Mpc) Determining the Age of the Universe v the velocity (v) of a galaxy at a certain distance (D) D From the Hubble Law H = can be determined if the Hubble Constant (H) is known and the law is assumed true. Velocity of a galaxy 800 Mpc away: 65978.4 km/s ♣ Converting mega parsecs to kilometres. (1 pc = 3.086 * 1013 km) 800 Mpc = 2.47 * 1022 The average relativistic value of the Hubble Constant determined previously was used. Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 9 To calculate the age of the Universe the relationship of v= D t Where v is velocity, D is distance and t is time was used. If the appropriate units are used and the equation rearranged then the age of the universe can be determined. Age of universe: 3.74 * 1017 s = 11.88 * 109 years The age of the universe is approximately 12 billion years old. Discussion The determination of the age of the universe is full of assumptions. The largest cause of error in the calculation of the Hubble Constant, and therefore the age of the universe, was the assumption that all galaxies have an absolute magnitude of –22. Galaxies range down to an absolute magnitude of approximately –12. It would have been better to classify the galaxies according to type and used varying absolute magnitudes for each type analysed (E0 → Irregular). An absolute magnitude of –21 results in an error of +59% in the distance. Using an absolute magnitude of -23 gives an error of –37%. These are quite substantial errors. Which do have an effect on future calculations and should be taken into account. Not all distant objects would have the same absolute magnitude. For example quasar, which are very distant objects (receding at 94.3% the speed of light!) would have an absolute magnitude many times that of a pulsar or neutron star that are comparably distant. A substantially shorter or longer integration time reduces the ability to discern the signal (spectra line) from the noise. It does not affect the wavelength that the line appears at. The limited number of galaxies used in the experiment has limited the accuracy of determining the Hubble Constant. A larger number should most certainly been used. The relativistic velocity calculation of the galaxies resulted in a slightly higher velocity than the Classical method and therefore a slightly younger age. Using the Hubble Constant derived from the classical calculation of velocity results in an age for Physics 210 The Age of the Universe Lab 3 Richard Laugesen Page 10 the universe of 12.7 billion years, where the relativistic derivation of the Hubble Constant gives an age of 11.9 billion years. At the edge of a galaxies’ disk the velocity would be different from the centre. Due to the rotation of the galaxy one side of the disk would be approaching and the other side would be receding. This local velocity would be superimposed upon the radial velocity of the galaxy as it recedes from us, so the redshift on one side of the galaxy would be slightly greater and on the other side it would be slightly less. This would only apply if the galaxy were orientated edge on to the earth.♥ Besides the errors due to assumptions made about absolute magnitude the experiment was a success. The calculated age of the universe is a little low. The ages of the oldest stars puts a lower limit of about 14 billion years on the age of the universe, radioactive isotope studies yield ages of at least 9 to 16 billion years for our galaxy. A reasonable lower limit would be about 15 billion years. The most recent study completed concerning the Hubble Constant gives a value of 70±7 km/s/mpc. This value in an Einstein-de Sitter universe results in an age of 10 billion years, and with the uncertainty in H0, a value of about 12 billion years is not out of the question for this model. But an age consistent with the implied lower limits would be closer to 15 billion years. References “Universe – Origin and Evolution”, Snow, 1997 “Lifting the Vail on Hubble’s Constant”, NASA, HTTP://science.nasa.gov/newhome/headlines/ast25may99_2.htm “Is Nothing Sacred”, John D. Barrow, New Scientist, July 1999 “Dynamic Astronomy”, Robert T. Dixon, 1984 This is a method to determine the angular velocity of a galaxy. Physics 210 The Age of the Universe Lab 3
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