explain how nuclear fusion in the Sun’s core releases energy, use E = mc² to connect the Sun’s luminosity to its mass loss, and describe the Sun’s life span from its birth to a white dwarf.
Instruction
The phenomenon. On January 24, 1848, James Marshall saw flakes of gold in the tailrace of a sawmill at Coloma, on the American River. Geologists can show that every atom of that gold is older than the Sun. By the end of this unit you will be able to explain how that can be true. We start with the star we know best.
What the Sun does. The Sun gives off about 3.83 × 1026 watts, its luminosity. At Earth’s distance (1.496 × 1011 m) that power is spread over a sphere, so each square meter facing the Sun receives L / (4πd2) ≈ 1,361 W. This is the solar constant, and it drives weather, climate and life.
Where the energy comes from. Chemical burning cannot explain it: a Sun made of coal would burn out in a few thousand years. The answer is nuclear fusion. In the core, at about 15 million K, hydrogen nuclei (protons) move so fast that they overcome their electric repulsion and fuse. Through a chain of steps, four hydrogen nuclei become one helium-4 nucleus. Four hydrogen atoms have a mass of 4 × 1.007825 = 4.031300 u; one helium atom has 4.002603 u. About 0.7% of the mass is missing, and it leaves as energy, E = mc2.
Worked example: how much mass does the Sun lose? Divide the power by c2: 3.828 × 1026 W / (2.998 × 108 m/s)2 ≈ 4.26 × 109 kg every second, about 4.3 million metric tons. Since only 0.7% of the fused mass disappears, the Sun fuses about 4.26 × 109 / 0.0071 ≈ 6 × 1011 kg of hydrogen per second. That sounds enormous, but the Sun’s mass is 2 × 1030 kg, so it can keep this up for billions of years.
From core to Earth. Energy released in the core moves outward first by radiation, as photons are absorbed and re-emitted countless times, then by convection in the outer third of the Sun, where hot gas rises and cool gas sinks. The energy leaves the surface (about 5,800 K) as light and takes only about 8.3 minutes to cross the 150 million km to Earth, although it spent many thousands of years working its way out of the interior.
The Sun’s life span. The Sun is about 4.6 billion years old and is roughly halfway through its hydrogen-burning life of about 10 billion years. It is stable because outward gas pressure from fusion heat balances inward gravity. As helium builds up in the core, the core contracts and heats, and the Sun slowly brightens. In about 5 billion years it will swell into a red giant, fuse helium into carbon and oxygen, shed its outer layers as a planetary nebula, and leave a white dwarf about the size of Earth. It is not massive enough to explode.
Your questions. A good scientific question can be investigated. “Why is the Sun hot?” becomes “What reaction could release this much energy for 4.6 billion years?” Your question board should hold questions like that about the Sun, the stars and the gold.
Formative check
Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.
Suppose the Sun were a giant ball of coal burning in oxygen. About how long could it shine at its present brightness?
Four hydrogen nuclei fuse into one helium nucleus. What happens to the 0.7% of mass that is ‘missing’?
The Sun converts about million metric tons of mass into energy every second, and its light takes about minutes to reach Earth.
Draw and label a model of the Sun that shows the core, the radiative zone, the convective zone and the surface, with arrows for how energy moves in each. Add two balanced arrows for gravity and gas pressure. Then write two investigable questions about the Sun or the Coloma gold for the class question board.
How confident are you that you can explain how fusion powers the Sun and describe the stages of its life?
Practice
Work these on paper or in your notebook, then open Check your answer. Aim for all of Fluency and Application; try at least one Challenge.
Printable version: this unit’s practice workbook (PDF)
Fluency
Build speed and accuracy with the core skill.
- Convert 1250 km/h to m/s.
Check your answer
Answer: 347.22 m/s - Convert 1250 kPa to atm.
Check your answer
Answer: 12.34 atm - Four hydrogen-1 atoms (1.007825 u each) fuse into one helium-4 atom (4.002603 u). What mass is ‘missing’, and what percentage of the starting mass is it?
Check your answer
Answer: 0.028697 u, about 0.71%4 × 1.007825 = 4.031300 u; 4.031300 − 4.002603 = 0.028697 u; 0.028697 / 4.031300 = 0.0071. - How much energy is released if 1.0 kg of mass is converted completely to energy? (c = 2.998 × 108 m/s)
Check your answer
Answer: 8.988 × 1016 JE = mc2 = 1.0 × (2.998 × 108)2. - Light travels 1.496 × 1011 m from the Sun to Earth at 2.998 × 108 m/s. How long does the trip take, in seconds and minutes?
Check your answer
Answer: about 499 s, or 8.3 minutest = d / v.
Application
Use the skill in context. Show your reasoning.
- The Sun’s luminosity is 3.828 × 1026 W. How much mass does it convert to energy each second, and each year (1 year = 3.156 × 107 s)?
Check your answer
Answer: about 4.26 × 109 kg/s; about 1.34 × 1017 kg per yearm = L / c2; multiply by the seconds in a year. - At that rate for 10 billion years, what fraction of the Sun’s mass (1.989 × 1030 kg) is converted to energy? Does this threaten the Sun’s stability?
Check your answer
Answer: about 0.07% of its mass; no, the Sun ends its main-sequence life because its core runs low on hydrogen, not because it runs out of mass4.259e+09 × 3.156 × 1017 s / 1.989 × 1030 = 6.76e-04. - The solar constant at Earth (1.000 AU) is 1,361 W/m2. Sunlight spreads as 1/d2. What is it at Mars (1.524 AU) and at Jupiter (5.203 AU)?
Check your answer
Answer: Mars about 586 W/m2; Jupiter about 50 W/m2Divide 1,361 by the distance in AU squared.
Challenge
Stretch problems. Expect to think before you write.
- Use the solar constant (1,361 W/m2) and Earth’s distance (1.496 × 1011 m) to calculate the Sun’s luminosity. Explain the geometry.
Check your answer
Answer: about 3.83 × 1026 WThe Sun’s power spreads over a sphere of radius d, area 4πd2; L = 1,361 × 4π(1.496 × 1011)2.
Review
Keep earlier skills sharp.
- What volume does 4.6 g of ice (density 0.917 g/cm3) occupy?
Check your answer
Answer: 5.0 cm3Volume = mass / density. - Convert 60 km/h to m/s.
Check your answer
Answer: 16.67 m/s