Lab 5: Understanding Stars through the H-R Diagram - Physics

of fuel, and you grill often enough to burn 5 lb per month, the tank will last for 4 months.For the sun, the fuel is the mass of hydrogen in its core (or more precisely the energy Mc 2of the mass). The rate it uses the fuel is given by the sun’s luminosity, L = 4 × 10 26 watts.The mass of the sun is M = 2×10 30 kg, and the mass in the core of the sun is about one tenththis total. Also, in the proton-proton cycle only about 0.007 of the mass of the protons isconverted to energy. After all, there are Helium nuclei left over after the reaction. Therefore,taking into account both factors, only 0.1 × 0.007 of the mass of the Sun is converted intoluminous energy. With this in mind, we have a formula for the lifetime of the Sun.lifetime of Sun = 0.0007 Mc2L .Recall the speed of light c = 3 × 10 8 m/s and plug in our data on M and L to getlifetime of Sun = 0.0007 2 × 1030 × (3 × 10 8 ) 2= 7 × 2 × 9 × 10 16 = × seconds.4 × 10 26 4The previous answer is difficult to comprehend in any meaningful manner. Convert it tobillions of years by dividing by the number of seconds in a billion years (roughly 3 × 10 16 s).The main sequence lifetime of the Sun =3.2 An AnalogyConsider two cars, A and B.billions of years.• Car A is a two-door coupe. It gets 50 miles per gallon and has a 10-gallon tank.• Car B is an SUV. It gets 2 miles per gallon and has a 30-gallon tank.1. Which car holds more fuel?2. Suppose both cars embark on a very long journey together, side-by-side, each with afull tank of gas. Which car will run out of fuel first?3. Explain the apparent contradiction between your previous two answers.4. Which car would you expect to put out more power? Explain your reasoning.3.3 Lifetimes and Stellar PropertiesAs you know from experience, it is possible for something to hold more fuel but run out ofthat fuel faster. This idea applies directly to stars. Since a star’s lifetime is longer if it has4

more mass (more fuel) but shorter if it has a higher luminosity (uses the fuel more quickly)a star’s lifetime is proportional to its mass divided by its luminosity, a quantity known asits mass-to-light ratio. More massive stars get crushed more under their own weight, leadingto higher temperatures, much faster fusion in their cores, and much higher power output, orluminosity. Similar to the SUV in the example, stars that are more massive and have morehydrogen will therefore have smaller mass-to-light ratios and always run out of hydrogen fuelin their cores sooner than those that are less massive. Put another way: stars that containmore mass live shorter lives than those containing less mass. The heavier a star, the shorterthe life span.Using the concepts given above, answer the following questions: Consider two mainsequencestars, Proxima Centauri (the closest star to the Sun) and Vega (the star from themovie Contact).• Proxima Centauri has a mass of 0.12 solar masses.• Vega has a mass of 2.6 solar masses1. Which star contains more hydrogen to fuse into helium?2. Which star will put out more power (i.e., use more energy per second and be moreluminous)? Explain your reasoning.3. Which star will run out of hydrogen first and thus leave the main sequence first?Generalize your result using the H-R diagram to tell you the relationship between the colorof a main-sequence star, its surface temperature, its luminosity, and its mass-to-light ratio.Fill in the blanks in the following table so that the relationships are expressed clearly.Temperature Luminosity Lifetime Mass-to-light ratioColor (hotter or cooler) (higher or lower) (longer or shorter) (higher or lower)BluerRedderNow let’s look at stellar lifetimes quantitatively. If a star had twice the mass of the Sunand the same luminosity, then it could be expected to last twice as long. On the other hand,if the star’s luminosity were four times greater than the Sun, it would not last as long. Thelifetime of a star depends directly on its mass-to-light ratio, and is approximately5

number of times more massive than Sunlifetime of star = × 10 billion years (1)number of times more luminous than SunAstronomers have measured the masses of a few of the main-sequence stars on your H-Rdiagram. The Table below lists these stars’ spectral types and approximate masses, togetherwith the mass and luminosity of a very hot, O-type star not on your diagram. Using yourH-R diagram, determine the luminosity of each of these stars. Then find the lifetimes of thestars using equation 1.Spectral Type Mass (M sun ) Luminosity (L sun ) Lifetime (years)O 50 500,000B8 5A 2K 0.5M 0.11. Explain why astronomers are unable to measure the masses of all the stars that arevisible. In what way does a star have to be special to allow its mass to be determined?2. Based on the table of lifetimes, try to figure out and describe reasoning that helpsexplain why O-type stars are so rare (and why there are none on your H-R diagram).3. (extra credit) As you will learn later in the course, the oldest stars in the Galaxy appearto be about 12 billion years old. Based on the table of lifetimes, try to figure out anddescribe reasoning that helps explain why white dwarfs are relatively rare even in thelist of the 20 closest stars, despite the fact that we think all main-sequence stars fromtype B on down and to the right (including the sun and every other main-sequencestar plotted on your HR diagram) become white dwarfs.6