Astronomy 20 HOMEWORK - Chapter 7 The Stars Use a calculator whenever necessary. For full credit, always show your work and explain how you got your answer in full, complete sentences on a separate sheet of paper. Be careful about units! Please CIRCLE or put a box around your final answer if it is numerical. If you wish, you may discuss the questions with friends, but please turn in your own hand-written solutions, with questions answered in your own way.. Chaisson Review and Discussion 7. How is parallax used to measure the distances to stars? (3 points) Parallax is the apparent change in position of a nearby object due to the change in the viewing position of the observer. For example, astronomers can view a nearby star over the course of six months, from opposite sides of the Earth s orbit. The position of this star appears to change, relative to the more distant stars in the background. The farther the star, the smaller the shift. The inverse of this motion measured in arc seconds is equal to the distance measured in parsecs. 2. Chaisson Review and Discussion 7.3 Explain two ways in which a star s real motion through space translates into motion that is observable from Earth. (4 points) A star s real space motion is observed as two components; the radial velocity and the transverse velocity. The radial velocity is just the star s motion towards or away from us; it can be determined using the Doppler Effect. The transverse motion is perpendicular to this motion, and can be determined by observing the star s proper motion, an angular motion among the other stars measured in seconds of arc per year. If the distance to the star is known proper motion can be converted into the true transverse velocity. To find the space velocity, the transverse velocity and radial velocity can be combined using the Pythagorean Theorem: (space velocity) 2 = (radial velocity) 2 + (transverse velocity) 2. 3. Chaisson Review and Discussion 7.4 How do astronomers go about measuring stellar luminosities? (3 points) The most straightforward way to find out two pieces of information about the star: its apparent brightness, and its distance. Once we know these things, we can find the absolute magnitude, a measure of how bright the star would appear if it were 0 parsecs away. With
the effects of distance removed, we can tell the star s luminosity. If distance is not known, we might be able to find luminosity from temperature and radius, if these are somehow known. 4. Chaisson Review and Discussion 7.5 Describe how astronomers measure stellar radii. (3 points) Most of the stars in the sky have no effective apparent size, even in our most powerful telescopes. Thus, direct determination of actual size is impossible. If the luminosity and temperature of a star are known, we can calculate the radius. Also, if two stars are eclipsing each other, we can measure their radii by timing how long it takes them to pass in front of each other. 5. Chaisson Review and Discussion 7.7 What is the difference between absolute and apparent brightness? (3 points) The apparent brightness of a star depends on two things: how bright it truly is, and its distance. It is easily measurable. The absolute brightness is a measure of the star s intrinsic brightness, or luminosity. Using the magnitude system, the absolute magnitude is the apparent magnitude the star would have if it were at a distance of 0 parsecs. For the absolute magnitude to be calculated, the apparent magnitude and distance to the star must be known. 6. Chaisson Review and Discussion 7.8 How do astronomers measure stellar temperatures? (3 points) If we can determine the wavelength of maximum emission of a star s radiation, we can find its temperature using Wien s Law. Also, we can use the B and V filters to gather and count blue and yellow photons, respectively. Comparing the amount of blue light to the amount of yellow light can tell us the star s temperature. 7. Chaisson Review and Discussion 7.9 Briefly describe how stars are classified according to their spectral characteristics. (4 points) The patterns of the dark lines in the spectrum of a star depend strongly on the star s temperature, since temperature determines which elements in the star s photosphere are capable of absorbing light and creating absorption lines. Spectra are classified according to temperature in the order O, B, A, F, G, K, M, with O-type stars being the hottest. Within each of these types is a numerical sub-classification ranging from 0 to 9, e.g. F0, F, F2,... F8, F9, G0, G,.... Within a specific spectral type, the number 0 is the hottest and 9 is the coolest. To classify a star s spectrum, we identify the sources of the spectral lines and compare the patterns to the standard classes. A spectrum with strong ionized helium lines is type O; one with strong hydrogen lines is type A; and so forth. 8. Chaisson Review and Discussion 7.0 Why do some stars have very few hydrogen lines in their spectra? (3 points)
Extreme temperatures on either end of the scale can cause weak hydrogen lines in a stellar spectrum. If the star is too hot most of the gas is ionized; without electrons to absorb light and make transitions from the second level, no hydrogen absorption lines will be created. If the star is too cool, all the electrons are in the ground state and unable to absorb visible light and create lines in the visible system. 9. Chaisson Review and Discussion 7.2 What is the main sequence? What is common to all stars on the main sequence? What basic property of a star determines where it lies on the main sequence? (4 points) About 90% of all stars plotted in the H-R diagram are found along a narrow S-shaped band running diagonally from upper left (hot and bright) to lower right (cool and dim). This is the main sequence. Stars along the main sequence all have a common source of energy, the fusion of hydrogen into helium, and all are in hydrostatic equilibrium. A star s position on the main sequence is uniquely determined by its mass: the most massive stars are in the upper left end while the lowest mass stars are in the lower right end. The Sun is in about the middle of the main sequence. 0. Chaisson Review and Discussion 7.3 How are distances determined by spectroscopic parallax? (3 points) The main sequence represents a consistent relationship between temperature and luminosity for main sequence stars. Thus, if we know that a star is a main sequence star (by seeing pressure broadening in the spectral lines), and we have some way of determining temperature (spectral type or color), we can place that star on the main sequence of the H-R Diagram. This will tell us the star s luminosity and absolute magnitude, which can then be combined with the apparent magnitude to find the distance.. Chaisson Review and Discussion 7.4 Why does the H-R diagram constructed from data on the brightest stars differ so much from the diagram constructed from data on the nearest stars? (3 points) The brightest stars in the sky also happen to be intrinsically very bright stars. Although seen at relatively large distances, they still appear bright. Therefore, the brightest stars will tend to be clustered in the upper part of the H-R diagram, mostly giants, supergiants, and large main sequence stars. The H-R diagram of the nearest stars is a more reflective of the galaxy as a whole: low mass stars dominate and the highest mass stars are rare. The lower part of the diagram is therefore quite full in such a diagram. 2. Chaisson Review and Discussion 7.7 How can stellar masses be determined by observing binary-star systems? (3 points) The mass of a star can be determined by observing its gravitational effect on an orbiting companion body. The combined mass of the system can be calculated using Kepler s third law, if the period of the orbit and the semi-major axis can be observed. If the location of the
center of mass of the system can also be determined, then the individual masses can be calculated. Star systems like visual and eclipsing binaries are very valuable assets in helping us find stellar masses. Spectroscopic binaries are helpful too, but can provide only partial information on the masses. 3. Chaisson Review and Discussion 7.9 In general, is it possible to determine the age of an individual star simply by noting its position on an H-R diagram? Explain. (3 points) High mass stars spend very short amounts of time on the main sequence. Therefore, if one sees a hot, bright, massive star on the main sequence, it has only recently arrived there, and must be very young. However, low mass stars use up their fuel very slowly. A low-mass, dim, cool star on the main sequence may have arrived there last year, or billions of years ago! 4. Jupiter is about 5 times as far from the sun as the earth is ( 5 A.U. s compared to A.U. ). By how much less is the sun s flux at Jupiter compared to that at the earth? (3 points) Flux = Luminosity / surface area or F = L / 4 r 2 Since the luminosity of the sun is constant for Jupiter and Earth we need only consider r in the above equation. Thus, FJupiter FEarth = rjupiter 2 = rearth 2 5 2 2 = 25 Fjupiter = 25 FEarth 5. The main-sequence star Regulus has a mass about five times that of the sun. Use the massluminosity relationship to estimate the luminosity of Regulus. (3 points) A star of five solar masses will be about 250x as luminous as the sun. The mass-luminosity relation yields L = M 3.5 = 5 3.5 = 279.5x as luminous as the sun. 6. The flux we receive from the Sun is about 370 Watts per square meter. (A watt is a unit of power which is equivalent to Joule per second). Sirius sends us a flux of about 0 7 W m 2. Luminosity is related to flux by the following equation: 2 L 4 r F
where L is the luminosity in Watts r is the distance in meters F is the flux in W/m 2 a) The Sun is.49598 x 0 m away, what is the Sun's luminosity? (3 points) L = 4 (.4958 x 0 m) 2 x 370 W/m 2 L = 3.85 x 0 26 W b) Sirius is 8.8 ly away ( ly = 9.46053 0 5 m). What is Sirius' luminosity? (3 points) L = 4 (8.8 ly x 9.46053 x 0 5 m/ly) 2 x x 0-7 W/m 2 L = 8.72 x 0 27 W c) If Sirius had a surface temperature of 5,000 K, what would its luminosity be then? We need to remember that flux has a temperature dependence through the equation... F T eff where T eff is the "effective" or surface temperature and is the Stefan-Boltzman constant. 56697. 0 8 W m 2 K 4. Assume Sirius has a radius of 8 0 8 meters - this is where the flux is measured. (5 points) F = Teff 4 = 5.6697 x 0-8 F = 2.87 x 0 9 W/m 2 L = 4 (8 x 0 8 m) 2 x 2.87 x 0 9 W/m 2 L = 2.3 X 0 28 W 4 W m 2 K 4 x (.5 x 04 K) 4 7. Stellar parallax allows us to find the distances to stars using simple trigonometry. The figure below shows the geometry.
Earth R p Sun d tan p R R so that d d tan p but the tangent of small angles is approximately equal to the angle itself in radians. So... d R, p if R is in A.U.'s and p is in arcsec's, then d is in parsec's and pc = 326. the smallest detectable parallax is about 0.00 arcsec; how far would a star with this parallax be? (3 points) ly. d = A.U. 0.00" = 000 pc 8. The relationship between the apparent and absolute magnitude of a given star is expressed as: m M 5log d 5 where m is the star's apparent magnitude, M is the star's absolute magnitude and d is the distance to the star in parsec's. If the star in problem # had an apparent magnitude of 5, what would its absolute magnitude be? (5 points) -M = 5logd 5 m M = -5logd + 5 + m M = -5log(000pc) + 5 + 5 M = -5(3) + 5 + 5 = -5
9. On the Hertzsprung-Russell (HR) diagram on the back of this page, plot points representing the Luminosity and Surface Temperature of each of the 6 brightest stars (in the northern hemisphere) on your list. (The values of luminosity (in units of the Sun's luminosity) and the surface temperature are given in your list of stars.) Write the name of each star next to its location on the HR diagram. (6 points) 20. Invent a new mnemonic for spectral classification. O-B-A-F-G-K-M (e.g. Oh boy, a furry gorilla kissed mom!). ( point) Oh Brother, Astronomy Formulas Gonna Kill Me!
HERTZSPRUNG RUSSELL DIAGRAM 0 6 0 5 Rigel Deneb Betelgeuse Antares 0 4 Spica 0 3 Regulus Capella Aldebaran Arcturus 0 2 Vega Castor Sirius Pollux 0 Fomalhaut Altair Procyon Sun 0-30 25 20 5 0 9 8 7 6 5 4 3 2 SURFACE TEMPERATURE (0 3 K)
6 BRIGHTEST STARS (VISIBLE FROM NORTH OF THE TROPICS) STAR DISTANCE APPARENT LUMINOSITY Tsurf( o K) TYPE (light years) BRIGHTNESS (Sun = ) (Compared to the sun if it were 32.6 ly distant. Sirius 8.7 30 38 9,700 (white main 2. Arcturus 36 86 200 4,600 (orange-red giant) 3. Vega 26.5 80 97 0,000 (white main 4. Capella 45 76 70 5,250 (yellow giant) 5. Rigel 900 74 20,000,700 (blue-white super giant) 6. Procyon.3 59 7.59 6,670 (yellow main Sequence) 7. Betelgeuse 520 57 0,000 3,00 (red super giant) 8. Altair 6.5 4 2 7,940 (white main 9. Aldebaran 68 39 500 4,50 (orange-red giant) 0. Spica 220 34 9,000 22,000 (blue-white main. Antares 520 35 20,000 3,200 (red super giant) 2. Pollux 35 29 53 4,850 (orange-red giant) 3. Fomalhaut 22.6 28 20 9,000 (white main 4. Deneb 600 26 0,000 9,200 (white super giant) 5. Regulus 87 24 40 3,000 (blue-white main 6. Castor 45 20 53 0,000 (white main