This homework asks you to do some simple calculations related to the production of energy in the Sun. Recall that the Sun produces energy by the fusion of hydrogen into helium in its core, and that in the process, it loses mass as some mass is converted into energy. The formula, Energy = Mass times (speed of light)² gives the conversion between mass and energy. Answer the following questions. [To get the right answer, it's important to express the quantities in consistent units. If you are uncertain about what units to use, refer to Appendix A of the textbook, especially page A8.]

1. How long will the Sun shine?   Recall that we are led to the idea of fusion in the Sun being it's energy source by the fact that no other energy source could power the Sun for its long history of 4.5 billion years. Will fusion do the job? You can calculate this as follows:
   
   The amount of mass available in the Sun to be used as "fuel" is the mass in the core of the Sun, which is approximately 10% of its mass. (The mass of the Sun is 2 x 10³⁰ kg.) Recall that, each time four hydrogen nuclei fuse into a helium nucleus, 0.7% of the mass of hydrogen is converted into energy (where E = mc²).
   
   You can put these numbers together to figure out the total amount of energy that the Sun can produce in its hydrogen-burning lifetime, i.e., as long as there is hydrogen available to "burn".
   
   Then, to figure out how long this lifetime is, you need to kow the rate at which the Sun is losing energy. We know this from the Sun's luminosity, which is 3.78 x 10²⁶ Joules/second (a Joule is a unit of energy, see page A8).
   
   Can you put this all together to figure out how long the Sun can shine (i.e., lose energy) at its present rate?
   
   
2. How does the time you've just calculated compare with the present age of the Sun? How much longer will the Sun shine as it does now?
   
   
3. We know that the Sun loses 3.78 x 10²⁶Joules of energy every second (this is the Sun's luminosity). By using E = mc², figure out how much mass this corresponds to. That is, how much mass does the Sun lose every second? Express your answer in metric tons. (A metric ton = 1000 kg.)
   
   
4. Sunspots look dark because they are at a temperature that is typically 1500 K cooler than the surrounding photosphere, whose temperature is 5780 K. Using the Stefan-Boltzmann law, compare the surface brightness of a dark sunspot to that of the surrounding photosphere. The surface brightness is the energy per unit time radiated from each square meter of the surface, either of the spot or of the photosphere.