Suppose a high-mass star starts out with a mass of 20.0 M, of which 75% is hydrogen (H) and 25% is helium (He). By the end of its life, the star has the structure shown in the diagram at right, with an iron (Fe) core of 1.4 M, surrounded by 0.6 M of sulfur (S), 1.5 M of oxygen (O), 2.5 M of helium, and 14.0 M of unburned fuel. 1. During the star's life, how much iron was produced by nuclear reactions? Give your answer in units of solar mass (M). The star produced just 1.4 M of iron. 2. How much sulfur was produced by nuclear reactions? (Hint: some of this sulfur was later `burned' to produce the iron.) Just 0.6 M of sulfur remains, while 1.4 M was burned to iron, so in total the star produced 2.0 M of sulfur. 3. How much oxygen was produced by nuclear reactions? Just 1.5 M of oxygen remains, while 2.0 M was burned to sulfur and iron, so in total the star produced 3.5 M of oxygen. 4. How much helium was produced by nuclear reactions in the star? (Hint: keep in mind the star's initial composition!) Just 2.5 M of helium remains, while 3.5 M was burned to oxygen, sulfur, and iron, requiring in total 6.0 M of helium. But of that total, 25% was present when the star formed, so the star produced only 75% of 6.0 M, or 4.5 M, of helium.

At each stage of nuclear burning, a tiny fraction of the mass is converted to energy. The table below lists the fractions converted at each stage in the sequence of nuclear reactions which take place in this star.

 Reaction 4H   ->   He 4He   ->   O 2O   ->   S 2S   ->   Fe Fraction 0.00717 0.00097 0.00055 0.00030

For example, when 1 M of iron is produced by the reaction   2S  ->  Fe  , just 0.00030 M of mass is converted to energy. (These fractions are small enough to be safely ignored in describing the star's final structure; that's why the star's final mass adds up to exactly 20.0 M).

5. During the star's life, how much mass was converted to energy by the reaction    2S  ->  Fe ?

1.4 M of iron was produced, yielding 1.4 M × 0.00030 = 0.00042 M worth of energy.

6. How much mass was converted to energy by the reaction    2O  ->  S ?

2.0 M of sulfur was produced, yielding 2.0 M × 0.00055 = 0.00110 M worth of energy.

7. How much mass was converted to energy by the reaction    4He  ->  O ?

3.5 M of oxygen was produced, yielding 3.5 M × 0.00097 = 0.00340 M worth of energy.

8. How much mass was converted to energy by the reaction    4H  ->  He ?

4.5 M of helium was produced, yielding 4.5 M × 0.00717 = 0.03227 M worth of energy.

9. Finally, when the iron core collapses, gravity converts 0.2 M of mass to energy. How does this compare to the total amount of mass converted to energy by nuclear reactions?

In total, 0.03718 M worth of energy was produced by nuclear reactions. This is only about 19% of the energy released by gravity. GRAVITY RULES!

Joshua E. Barnes (barnes@ifa.hawaii.edu)