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Unformatted text preview: PHYS 302: Unit 14 Viewing Notes – Athabasca University PHYS 302: Vibrations and Waves – Unit 14 Viewing Notes The initial discussion about whistles should be familiar (e.g., train whistles). The perceived change in pitch due to motion is called a Doppler shift. In acoustics, the speed of the medium, the sound source, and the participant all play a role in determining the perceived pitch. Whether the observer or the source is moving with respect to the medium is significant. Note that the situation presented is asymmetric. Recall that the speed of sound is about 340 m/s (memorize this). Dr. Lewin shows without proof that the received frequency f’ at a receiver moving with speed vr from a transmitter vs vr vs vt of frequency f moving at speed vt is f ' f , where vs is the speed of sound (3:12). If vr is the sound speed vs, then the receiver is moving at the speed of sound away from the transmitter, and the received frequency becomes zero: the receiver doesn’t receive any sound, because the sound cannot catch up to the receiver. If the sound transmitter is moving away at this speed, then vt=-vs and the formula gives f ' 1 f (4:15). There is a huge 2 difference between these cases even though the relative motion of the source and the observer is the same. In the demo, a 4000 Hz tuning fork moved about 1 m/s shows a variation from about 4012 to 3988 Hz, which is heard readily enough (6:00). Rather than moving in a straight line, sources of waves often travel on a circular path with speed around the circumference v0 R . When the source is moving perpendicular to the line of sight, the frequency stays the same; the frequency increases when the source is moving toward the observer; and it decreases when it is moving away from the observer (as it moves around the circle). The component of velocity moving in the observer’s direction is called radial velocity. When the source is farthest away from the observer and moving transversely, the radial velocity is vrad v0 sin t if the time zero is chosen (8:00). A lot can be deduced about the source’s motion from observations leading to the determination of the radial velocity. One can deduce the period T, and from the extrema f 'max and f 'min one can get v0 and the radius of motion R. Dr. Lewin demonstrates this with a whirling whistle (9:40). Electromagnetic radiation also shows a Doppler effect, with the same basic phenomenology. Derivation requires special relativity, and the velocities are no longer absolute. (Note, in acoustics, the medium can be used as a frame of reference). Without proof, the Doppler formula for EM radiation is expressed in terms of wavelength as ' 1 cos 1 2 , where v . For 0 º <θ<90º, the object is approaching the observer, λ’<λ, and the light is c bluer than it would be if it was not moving (higher frequency), so it is called a blue shift. For 90º<θ<180º, the object is receding from the observer, λ’>λ, and the light reddens (lower frequency), so it is called a red shift (14:15). These terms are commonly used in astronomy. If β2 is significantly less than 1, the denominator is about 1, so the simpler form ' (1 cos ) can be used. Radar works by emitting a known wavelength and assessing the reflected wavelength received. Stars show absorption lines at known wavelengths, so measuring these lines allows for easy determination of the radial velocity of stars. For example, Delta Leporis has Ca absorption (similar to the Na absorption demonstrated earlier) with a known wavelength (at rest) of λ=3933.664 Å (one Å or Ångstrom is 0.1 nm, or 10-10 m), but the observed wavelength is λ’=λ+1.298Å. From this, vcosθ is 99 km/s (relative) (19:15). In astronomy, the ability to measure Doppler shifts is important, and Dr. Lewin examines two examples. First, he discusses black hole binaries. Many stars are binary (two stars appear to orbit around a common centre of mass), but sometimes one cannot be seen. Nevertheless, the absorption lines in the visible star will show Doppler shifts as the stars orbit each other. The period T can be derived from these absorption lines. If both stars are detectable, there are 1 PHYS 302: Unit 14 Viewing Notes – Athabasca University two sets of lines, and it is easy to determine the orbit’s radius. In the case presented, only one set of lines is visible. In such a system (for example where one object is a black hole), the two objects must be on opposite sides of the centre of mass, so that m1r1=m2r2. Assuming that the orbits are circular, simple algebra and T lead to radii r1 and r2, and speeds v1 and v2. From Kepler’s Third Law (which you may have studied before) the period, radii, and masses are related by T2 4 2 (r1 r2 )3 , where G is the universal gravitational constant (value 6.67300 × 10-11 m3 kg-1 s-2). (m1 m2 )G (24:50). This equation allows us to determine the sum of the masses of the stars, and with the centre of mass formula, m1r1=m2r2, one can solve for the individual masses. All of this assumes that one is in the plane of the orbit. If the orbit is tilted so that we do not see it edge-on, the radial velocity will be lower. Eclipsing binary systems must be edge-on, so this is not a problem here. Solving for the tilt in the general case is not easy, and often it cannot be done. It is not discussed further here. Some binary systems consist of an ordinary star with a neutron star or black hole close to it. The attraction of each star is equal at a point called the Inner Lagrange point. If this point is inside the bigger star, matter will stream from it to the companion in a spiral pattern due to orbital motion. The star losing mass is called the donor, and the star acquiring mass is called the accretor. The spiralling material forms an accretion disk. Calculating the energy that will arrive at the donor with the material is a question of elementary mechanics. When potential energy becomes kinetic energy, we get about mMG 1 2 mv or v R 2 2MG , which is also known as the escape velocity (30:20). R The mass of a neutron star is about 1.5 times that of the Sun, but the radius of a neutron star is tiny—only 10 km. The energy associated with matter falling into such a potential well is enormous, so infalling material gets very hot and radiates mainly X-rays. Even a marshmallow falling in would release the energy equivalent of an atomic bomb (32:00). Often, the mass of an accretor is about 1.4 times that of the Sun, which is the Chandrasekhar limit on stability of white dwarf (planet-sized) stars, and this mass is also typical for a neutron star. The mass limit of a neutron star is about three solar masses. If a neutron star exceeds three solar masses, it should turn into a black hole. The gravity of a black hole is so strong that light cannot escape: the escape velocity is greater than the speed of light. At the event horizon, the escape velocity is c: c 2MG . If the Earth was compressed to be a black hole, the R radius of Earth would be 1 cm (rather than 6400 km); for the Sun, it would be 3 km; and for five solar masses, it would be 15 km (scaling linearly with mass) (40:00). In 1971, Bolton and others concluded that the double star X-ray source Cygnus X-1 must have a black hole in it (43:00). Break In the 1920s, astronomers measured the radial velocities of many stars, finding radial velocities up to a few hundred km/s (typically). Some nebulae in the skies had large red shifts (over 1000 km/s). Eventually astronomers realized that these were galaxies with about 10 billion stars each that showed an average spectrum from these stars. Hubble discovered a correlation between the distance of galaxies from Earth and their redshift. The distances, found in a complex way using variable stars, were underestimated initially. The linear proportionality of distance and redshift is now known as Hubble’s Law: v Hd , with H being about 70 km/s/Mpc, where an Mpc is 3.26×106 light years, or 3.1×1019 km. The dimension is actually [1/s] or [1/time]. The proportionality of distance and redshift allows for the calculation of distances using measured redshifts and also the estimation of how long ago light left a galaxy. In the examples (pictures) of galaxies that Dr. Lewin shows (50:00), those at a larger distance from Earth appear smaller, and the redshift of absorption lines shifts more towards the red. Comparison laboratory spectra are shown on the edges of the observed galaxy spectra. 2 PHYS 302: Unit 14 Viewing Notes – Athabasca University Early in the 21st century, Hubble Space Telescope data led to the most accurate value for Hubble’s constant, about 72 km/s/Mpc (concluded by Wendy Freedman). Hubble’s original data would not even be easily seen on the graph used to determine this constant. Despite all of the galaxies that appear to recede from Earth and the inference of an initial giant explosion called the Big Bang, we cannot conclude that Earth is at the centre of the universe. However, we can estimate the age of the universe simply through d=vtu. With v=Hd, d=Hdtu, that is, Htu=1, or tu=1/H. Taking the value of H and doing unit conversion, we can estimate the age of the universe at tu=4.3×1017s or 14 billion years. The oldest stars are about 10 billion years old. Note that the subject of deceleration is under active study. Redshift can lead to shifts in the entire wavelength range in which a line appears. For example, UV wavelengths can become visible if the redshift is high enough (1:30:30). Does all this, in fact, appear to put Earth at the centre of the universe? Consider an expanding loaf of raisin bread in an oven. From the perspective of any single raisin, the others all seem to be moving away, and the speed of movement will depend on the distance of each raisin from the observation point (Hubble’s Law for raisin bread). We can draw a similar analogy for the surface of an expanding balloon, if distances are measured solely along the surface (demo: 1:07:00). The remainder of the lecture discusses the changes of cosmological aspects of the universe with time—a subject of active research. You should recognize that cosmological redshifts actually arise due to expansion of the universe, including the local space in which they are measured. While the mathematics are the same as that for the Doppler shift, these changes go beyond simple relative motion. In our analysis (above), we did not in fact derive the Doppler shift for light; we simply presented it as a result of special relativity. The analysis needed to derive the equations for gravitational redshift correctly falls in the domain of general relativity, which is well beyond what we cover in this course. 3 ...
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