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PHYSICS 1311 Lab Exercise #9 Plotting the H-R Diagram

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PHYSICS 1311 Lab Exercise #9 Plotting the H-R Diagram Introduction The development of astronomical instruments accurate enough to permit measurement of parallax angles for nearby stars opened the door for one of astronomy's most significant developments, a development that, for the first time, yielded some real understanding about the nature of stars. There are two properties of stars that we will use in this exercise - the absolute magnitude, or luminosity, and surface temperature of stars. The surface temperature is measured in Kelvins and is straightforward. Absolute magnitude is a measure of the luminosity of a star, or the amount of energy it radiates. When Henry Norris Russell and Einar Hertzsprung first plotted these two properties of nearby stars early in the 20th century, they produced a diagram known to this day as the Hertzsprung-Russell Diagram, or H-R Diagram for short. It was a revolution in the understanding of stars. To define absolute magnitude, we start with a star's apparent magnitude, or the brightness of the star as it appears in the night sky. Some stars are obviously bright and many others are faint. The ancient Greeks devised a simple system of ranking stars according to brightness; our current system has its origins in that ancient system. The Greeks defined 6 levels of brightness, with 1 being the brightest and 6 being the faintest. When instruments capable of measuring star brightness were developed, astronomers found that a magnitude 1 star was about 100 times brighter than a magnitude 6 star. This was convenient, so this brightness difference was then defined as exactly 100. In the current system, a magnitude 1.0 star is exactly 100 times as bright as a magnitude 6.0 star. This is a 5 magnitude brightness difference. A one magnitude difference then becomes the fifth root of 100, or 2.512. A magnitude 1.0 star is 2.512 times brighter than a magnitude 2.0 star, 6.31 (2.5122) times brighter than a magnitude 3.0 star, 15.85 (2.5123) times brighter than a magnitude 4.0 star and so on. The following table lists some factors. m2 - m1 b1/b2 1.0 2.51 2.0 6.31 3.0 15.85 4.0 39.81 5.0 100.0 6.0 251.2 7.0 631 8.0 1585 9.0 3981 10.0 10000 9-1 Apparent brightness of a star is represented by "m". The luminosity difference between any two stars is simply 2.512 raised to a power equal to the magnitude difference. For example, if the magnitude difference between 2 stars is 10.5, then the luminosity difference is 2.51210.5. Absolute magnitude is defined as the apparent brightness in our night sky that a star would have if it could somehow be placed at a distance of exactly 10 parsecs (32.6) light years from Earth. This value tells you the luminosity of the star. The apparent magnitude of the star does not by itself tell you how luminous the star actually is unless you know the distance to the star. Without the distance you know nothing about the luminosity, because apparent magnitude varies with both luminosity (absolute magnitude) and distance. The first distance measurements were made on stars close enough for parallax measurements to be made. This work yielded the distances to a number of stars relatively close to the Sun, less than 50 light years away. With these distances, absolute magnitudes could then be calculated as follows. The distance, absolute magnitude and apparent magnitude of a star are related according the following simple formula. d m−M=5log( ) 10 where m is apparent magnitude, M is absolute magnitude and d is distance in parsecs. The number 5 comes from the relation of 5 magnitudes being a brightness ratio of 100. The 10 in the denominator of the fraction is the 10 parsecs for the standard distance. It is very important to note that if you know the values of any two of the parameters m, M, and d, then you can solve this equation for the third value. The equation can be rearranged as follows. m−M=5 log(d)−5 Here is the formula rearranged three more ways so it can be solved for each of the three quantities given the other two. d M=m−5log( ) 10 d m=M +5log( ) 10 m−M d=10(10 5 ) Now for an example. Suppose we look at Alpha Centauri A, which is the brighter 9-2 component of the bright star Rigil Kentaurus in the southern sky. Its apparent magnitude m is 0.0, which is quite bright. This brightness is, however, due to its being relatively close as stars go. If you could transport it from its distance of about 4.2 light years (1.289 parsecs) away to 32.6 light years (10 parsecs) away, its brightness would fall to magnitude 4.4. Plugging this into the formula yields the following. 1.29 0.0−4.4=5 log( ) 10 The two sides are equal (within limits of the example's approximations). The luminosity of a star is a function of its temperature and its size. First, we need to know how much energy one square centimeter of star radiates per second. This is expressed in Stefan's Law as follows. Energy emitted 4 =σT cm2 second where σ equals 5.67 x 10-7 to produce a metric result. Energy radiated per square centimeter rises as the fourth power of the temperature. As an example, consider two stars - one with a temperature of 5,000 C and another with a temperature of 20,000 C. The hotter star is 4 times hotter than the cooler one, but the hotter one emits 256 times more energy per square centimeter. The total energy output, or luminosity, of a star is the energy output value for one square centimeter multiplied by the total surface area of the star in square centimeters. L=4π R2 σ T 4 This means that if you have two stars with the same surface temperature, the larger one will be more luminous because it has more surface area to radiate. Suppose you were to compare the luminosities of two stars with the same effective temperature. You would divide as follows. Note that A is a star’s surface area. 4 L1 A1 σ T = 4 L2 A2 σ T Canceling the equal terms above and below leaves only the ratio of the radii squared. Remember that the temperature is the same for both stars, so it cancels out. L A 1 = 1 L2 A2 The luminosity ratio is simply equal to the surface area ratio. Just remember that the area ratio is equal to the square of the radius ratio. You double the radius of a sphere and get 4 times the surface area. 9-3 Objectives 1. To plot real data about a number of stars and discover the structure in the physical parameters of stars. 2. To examine the Main Sequence of stars. Equipment Stellar data (printed below), graph paper, pencil. Procedure 1. Make your plots on the graph page in the data sheets. There is a small mark in the graph at the position for the Sun. 2. Plot the data from Table 1 on the paper. Make a small dot mark on the graph paper at the point that corresponds to the Absolute Magnitude and the surface temperature. Don't enter the star names on the graph – there’s no way they will fit. 3. Plot the data from Table 2 on the paper. Make a small circle mark on the graph paper at the point that corresponds to the Absolute Magnitude and the surface temperature. Be certain that you can distinguish these marks from those of part 4 above. Once again, don't enter the star names on the graph. The nearest stars are all located within a sphere of space about 15 light years in radius. 9-4 TABLE 1: The Nearest Stars Star Absolute Spectral Surface Temp Magnitude Type (Kelvins) Sun 4.85 G2 5,600 Proxima 15.45 M5.5 2,600 Alpha Centauri B 5.7 K1 4,000 Alpha Centauri A 4.34 G2 5,800 Barnard's Star 13.24 M5 2,600 Wolf 359 16.57 M6.5 2,400 BD +36 2147 10.46 M2 2700 Sirius A 1.45 A1 9,500 Sirius B 11.33 A2 28,000 Luyten 726-8 A 15.4 M5.5 2,500 Luyten 726-8 B 15.8 M5.5 2,400 Ross 154 13.0 M3.6 2,650 Ross 248 14.77 M5.5 2,500 Epsilon Eridani 6.2 K2 4,500 CD -36 15693 9.76 M2 2,950 Ross 128 13.5 M4 2,600 61 Cygni A 7.5 K5 4,000 61 Cygni B 8.3 K7 3,700 Procyon A 2.7 F5 6,500 Procyon B 13.0 F0 7,000 BD +59 1915 A 11.1 M3.5 2,650 BD +59 1915 B 11.9 M4 2,600 BD +43 44 A 10.3 M2 2,950 BD +43 44 B 13.2 M4 2,700 G51-15 17.0 M6.5 2,100 Epsilon Indi 6.9 K4 4,000 Tau Ceti 5.7 G8 5,000 L372-58 15.21 M4.5 2,650 L725-32 14.25 M5.5 2,500 BD +5 1668 11.9 M4 2,700 Kapteyn's Star 10.9 M1 3,300 CD -39 14192 8.7 M0 3,300 Kruger 60 A 11.6 M3.5 2,900 Kruger 60 B 13.3 M4 2,650 Ross 614 A 13.1 M4 2,650 BD -12 4253 12.0 M4 2,650 CD -37 15492 10.3 M2 2,900 Wolf 424 A 14.4 M5 2,500 van Maanen's Star 14.3 G0 5,800 L 1159-16 14.0 M4.5 2,650 L143-23 15.6 M4 2,650 CD-25 10553B 13.8 M1.5 3,200 BD+68 946 10.87 M3.5 2,900 LP731-58 17.3 M6.5 2,100 CD -46 11540 11.0 M3 2,700 G208-45 15.7 M6 2,200 G 158-27 15.4 M5 2,500 BD-15 6290 11.8 M5 2,500 9-5 TABLE 2: The Brightest Stars Canopus -5.4 A9 6,400 Arcturus -0.6 K1.5 3,900 Vega 0.6 A0 9,700 Capella A -0.7 G6 5,000 Rigel A -6.8 B8 11,000 Betelgeuse -7.2 M2 2,700 Achernar -1.0 B3 13,500 Beta Centauri -4.0 B0.5 20,000 Altair 2.1 A7 7,700 Alpha Crucis A -4.0 B0.5 19,500 Alpha Crucis B -3.6 B1 16,500 Aldebaran A -0.8 K5 3,500 Spica -3.6 B1 19,500 Antares A -5.8 M1.5 2,700 Pollux 0.8 K0 4,100 Fomalhaut A 1.6 A3 8,900 Deneb -7.5 A2 9,400 Beta Crucis -4.6 B0.5 20,500 Regulus -0.3 B7 11,500 4.
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