Stellar Evolution: PECTAA Class 12 Physics, Punjab Board

Every star is a fight. Gravity pulls the gas in. Pressure pushes it out. While the two are equal, the star shines. When one wins, the star changes.

Section 13.6 of Chapter 13 (Thermal Physics) in the PECTAA Physics 12 book is Stellar Evolution. This note covers it for FSc Part 2 (2nd year, Class 12) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

Birth of a star. The book says all stars form from clouds of gas and dust called nebulae or molecular clouds. As the cloud contracts, its density and temperature rise because the kinetic energy of the particles rises (K.E. is proportional to T). The contracting cloud is the proto-star. After millions of years it can reach a state of equilibrium and become a star.

Lifetime. Star lifetimes range from a few million years to trillions of years. The life of a star depends on its mass.

Balance of forces. As temperature rises, the internal pressure rises. A stable star forms when the inward gravitational force is exactly balanced by the outward pressure. The book writes this as the hydrostatic equation:

ΔP / Δr = - G M_r ρ_r / r²

• P is the pressure inside the star (force per unit area).
• ΔP/Δr is how the pressure changes from the centre outward.
• r is the distance from the centre.
• G is the gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻².
• M_r is the mass enclosed within radius r.
• ρ_r is the density at radius r.
• The negative sign shows that pressure decreases outward while gravity pulls inward.

The book lists two cases if the balance is lost: gravitational force greater than internal pressure, or gravitational force less than internal pressure.

Energy source. When the core temperature rises enough, nuclear fusion starts. Hydrogen nuclei combine to form helium. This releases a huge amount of energy and keeps the star in equilibrium for most of its life.

Red giant. With more heating and changes in the core, the star may expand to become a red giant. Stars with at least about half the mass of the Sun can start helium fusion in the core, making heavier elements such as carbon and oxygen. In more massive stars, successive stages of fusion form heavier nuclei up to iron.

Final fate. It depends primarily on mass.

• Low and intermediate mass stars: the core contracts into a dense white dwarf and the outer layers are expelled as a planetary nebula.
• Stars roughly ten times the mass of the Sun: a supernova explosion. The inert iron core collapses and leaves a dense remnant, a neutron star or a black hole.
• Red dwarfs: low mass and highly efficient use of fuel, so a much slower life. The book says the universe is not yet old enough for any red dwarf to have finished its life cycle.

Neutron stars. They are among the densest known objects, second only to black holes. Matter inside behaves like a degenerate gas. Young, fast-rotating neutron stars that emit beams of radiation are pulsars. The book says one teaspoon of neutron star material would have a mass over 5.5 × 10¹² kg, and the whole Earth at this density would fit in a sphere 305 m across.

The Sun. It formed about 4.6 billion years ago from a solar nebula. Gravitational contraction raised the core temperature and made a protostar. Fusion of hydrogen into helium then began. Today the Sun is in the main sequence stage.

Key points

Gravitational pressure. The book examples use the approximation P_g ≈ G M² / R⁴, where M is the mass and R the radius of the star. Compare P_g with the internal pressure:

• P_g close to internal pressure: stable equilibrium.
• P_g greater than internal pressure: the star contracts and may collapse.
• Internal pressure greater than P_g: the star expands and may become a red giant.

Worked examples

Example 1 (book Example 13.5). A star has mass 1 × 10³⁰ kg, radius 1 × 10⁹ m and internal pressure 5 × 10¹⁴ Pa. Find its state.

1. P_g = (6.67 × 10⁻¹¹)(1 × 10³⁰)² / (1 × 10⁹)⁴.
2. = 6.67 × 10⁴⁹ / 10³⁶ = 6.67 × 10¹³ Pa.
3. Internal pressure (5 × 10¹⁴ Pa) is larger, so the star expands and may become a red giant.

Example 2. A star has mass 3 × 10³⁰ kg, radius 1 × 10⁹ m and internal pressure 5 × 10¹⁴ Pa. (Worked with the book formula.)

1. P_g = (6.67 × 10⁻¹¹)(3 × 10³⁰)² / (1 × 10⁹)⁴.
2. = 6.67 × 10⁻¹¹ × 9 × 10⁶⁰ / 10³⁶ = 6.0 × 10¹⁴ Pa.
3. P_g is larger than 5 × 10¹⁴ Pa, so gravity is stronger and the star contracts.

Example 3. Name the path of a star like the Sun, using the book list.

1. Nebula, then protostar.
2. Main sequence, where it makes heat and light by converting hydrogen into helium.
3. Red giant, planetary nebula, white dwarf.

Example 4. A stellar core of mass about ten times the Sun's collapses after iron forms. What are the possible remnants?

1. The book says the inert iron core collapses in a supernova.
2. The remnant is a neutron star or a black hole.

Common mistakes in the exam

1. Dropping the negative sign. It shows pressure falls outward while gravity pulls inward.
2. Saying every star fuses up to iron. The book says that happens in more massive stars.
3. Mixing the fates. White dwarf is for low and intermediate mass. Supernova is for about ten solar masses.
4. Errors in powers of ten. Write the powers of ten line by line when using P_g = G M² / R⁴, and recheck each step.
5. Comparing the wrong way. If P_g is larger, the star contracts. If internal pressure is larger, it expands.
6. Calling a pulsar a different object. It is a rapidly rotating neutron star.

Exam technique

• Draw a small diagram with an arrow inward for gravity and one outward for pressure.
• Define each symbol of the hydrostatic equation.
• For a numerical, write the formula, substitute, then state the comparison in a sentence.
• Check your board's notice for the sections in your paper.

Practice questions

1. Write the hydrostatic equation and define M_r. Answer: ΔP/Δr = - G M_r ρ_r / r². M_r is the mass enclosed within radius r.
2. Which fusion process powers a star for most of its life? Answer: Hydrogen nuclei combine to form helium.
3. Name the two possible remnants of a supernova. Answer: A neutron star or a black hole.
4. A star has M = 1 × 10³⁰ kg and R = 1 × 10⁹ m. Find P_g. Answer: P_g = G M² / R⁴ = 6.67 × 10¹³ Pa.
5. What are pulsars? Answer: Young, rapidly rotating neutron stars that emit beams of radiation.

Ready to practise this? Continue in 2nd Year Physics.