||PRINT Name   __________________________
|Fall 2005   Section 006
: Sunshine & Sunspots
Thursday, Oct 27, 2005)
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.]
- 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
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
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?
- 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?
- 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.)
- 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.