Stellar evolution: a lifelong tug of war

The problem

The Sun is a ball of gas about 7 × 10⁸ m across, with a mass of about 2 × 10³⁰ kg. Its own gravity pulls every part of it towards the centre. So why hasn't it collapsed into a tiny dot?

Because the hot gas inside pushes back. Kinetic theory says hot gas means fast molecules and high pressure. A star's whole life is a tug of war between gravity pulling in and pressure pushing out.

A stable star is in hydrostatic equilibrium: the inward pull of gravity is exactly balanced by the outward pressure, ΔP/Δr = −GMᵣρᵣ/r². If gravity wins the star contracts; if pressure wins it expands.

Key terms

Nebula to protostar

Stars form from clouds of gas and dust called nebulae or molecular clouds. As a cloud contracts, its density and temperature rise, because the particles gain kinetic energy. This is a protostar.

Main sequence

When the core is hot enough, nuclear fusion turns hydrogen into helium and releases immense energy. This holds the star up against gravity for most of its life. The Sun is in this stage now.

Red giant

Rising core temperature makes the star expand greatly into a red giant. Stars of at least about half the Sun's mass then fuse helium into carbon and oxygen; massive stars go on fusing heavier nuclei, up to iron.

The end depends on mass

Low- and intermediate-mass stars shed their outer layers as a planetary nebula and leave a white dwarf. Stars about ten times the Sun's mass explode as a supernova, leaving a neutron star or a black hole.

Short questions with model answers

1. Q1. A star has mass 2 × 10³⁰ kg and radius 7 × 10⁸ m. Estimate its gravitational pressure and compare it with an internal pressure of 1.0 × 10¹⁵ Pa.

   • P_g ≈ GM² ÷ R⁴
   • P_g = 6.67 × 10⁻¹¹ × (2 × 10³⁰)² ÷ (7 × 10⁸)⁴ = 2.67 × 10⁵⁰ ÷ 2.40 × 10³⁵
   • P_g ≈ 1.1 × 10¹⁵ Pa ≈ P_internal

   About 1.1 × 10¹⁵ Pa, nearly equal to the internal pressure: the star is stable.

2. Q2. A massive star has mass 4 × 10³⁰ kg, radius 5 × 10⁸ m and internal pressure 1 × 10¹⁶ Pa. Is it stable?

   • P_g = 6.67 × 10⁻¹¹ × 16 × 10⁶⁰ ÷ 6.25 × 10³⁴ ≈ 1.7 × 10¹⁶ Pa
   • 1.7 × 10¹⁶ Pa > 1 × 10¹⁶ Pa

   Gravity wins: the star contracts and may collapse into a neutron star or black hole.

3. Q3. A star has mass 1 × 10³⁰ kg, radius 1 × 10⁹ m and internal pressure 5 × 10¹⁴ Pa. Determine the state of the star.

   • P_g = 6.67 × 10⁻¹¹ × 10⁶⁰ ÷ 10³⁶ = 6.67 × 10¹³ Pa
   • 5 × 10¹⁴ Pa > 6.67 × 10¹³ Pa

   Internal pressure wins: the star expands and may evolve into a red giant.

Common mistakes

• ✗ Writing that the most massive stars live the longest, as one sentence in the book says.

  ✓ It is the reverse. Massive stars burn their fuel fast and live only a few million years. Red dwarfs, the least massive, outlive the current age of the universe, as the book says on the next page.

• ✗ Saying a star expands when its gravity is greater than its internal pressure.

  ✓ When gravity wins, the star contracts and its core temperature increases (MCQ 13.8). It expands only when the internal pressure wins.

• ✗ Writing that every star ends as a black hole.

  ✓ The end depends on mass: low and intermediate mass → white dwarf; about ten solar masses or more → supernova, then a neutron star or a black hole.

MCQs

1. 1. Which equation ensures that a star remains stable against gravitational collapse?

   1. (a) mass continuity equation
   2. (b) energy gravitational equation
   3. (c) hydrostatic equilibrium equation
   4. (d) energy transport equation
   Show answer

   (c) Hydrostatic equilibrium: ΔP/Δr = −GMᵣρᵣ/r², pressure balancing gravity (MCQ 13.7).

2. 2. What happens to a star when its gravitational force becomes greater than its internal pressure?

   1. (a) it expands and becomes a red giant
   2. (b) it contracts and its core temperature increases
   3. (c) it remains stable with no changes
   4. (d) nuclear fusion stops immediately
   Show answer

   (b) Gravity wins, so the star contracts; the squeezed gas gets hotter (MCQ 13.8).

3. 3. In which stage of life is the Sun at present?

   1. (a) protostar
   2. (b) main sequence
   3. (c) red giant
   4. (d) white dwarf
   Show answer

   (b) Main sequence: it produces heat and light by converting hydrogen into helium.

4. 4. Stars with about ten times the Sun's mass end their lives as:

   1. (a) a planetary nebula only
   2. (b) a white dwarf
   3. (c) a supernova, leaving a neutron star or black hole
   4. (d) a red dwarf
   Show answer

   (c) Their iron core collapses in a supernova, leaving a neutron star or a black hole.

5. 5. Which stars live the longest?

   1. (a) the most massive stars
   2. (b) red dwarfs
   3. (c) red giants
   4. (d) neutron stars
   Show answer

   (b) Red dwarfs: low mass and efficient fuel use. None has yet finished its life, because the universe is not old enough.

Quick revision

• A star is stable in hydrostatic equilibrium; gravity winning makes it contract, pressure winning makes it expand.
• Life cycle: nebula → protostar → main sequence → red giant → white dwarf, or supernova → neutron star or black hole for massive stars.
• Massive stars live fastest; red dwarfs live longest. The Sun, 4.6 billion years old, is on the main sequence.