E.5.2 Evolution of stars Notes

Stellar Stability, Evolution, and End States

1. Stars are engaged in a constant tug-of-war between two opposing forces: gravity and radiation pressure.
2. Gravity, driven by the star’s immense mass, pulls it inward, seeking to collapse it.
3. Meanwhile, nuclear fusion in the star’s core generates energy that creates an outward radiation pressure, pushing against gravity.
4. When these forces are perfectly balanced, the star remains stable.

Why Does Fusion Create Outward Pressure?

1. At the core of a star, hydrogen nuclei (protons) fuse to form helium through nuclear fusion.
2. This process releases energy because the mass of the resulting helium nucleus is slightly less than the combined mass of the hydrogen nuclei.
3. The lost mass is converted into energy according to Einstein’s equation: $$E = \Delta m c^2$$
4. This energy radiates outward, generating pressure that counteracts the inward pull of gravity.
5. Stable stars maintain equilibrium through the balance of inward gravitational pressure and outward radiation pressure generated by nuclear fusion.

Classifying Stars: The Hertzsprung-Russell (HR) Diagram

1. The HR diagram is one of the most important tools in astrophysics.
2. It plots luminosity (brightness) on the vertical axis and surface temperature on the horizontal axis, with temperature increasing to the left.

Main Features of the HR Diagram

1. Main Sequence:
   1. A diagonal band where most stars, including the Sun, are found.
   2. These stars are fusing hydrogen into helium in their cores.
2. Red Giants and Supergiants:
   1. Found in the upper-right region of the diagram.
   2. These stars are large, cool, and luminous.
3. White Dwarfs:
   1. Located in the lower-left region.
   2. These stars are small, hot, and faint.

What Can the HR Diagram Tell Us?

1. A star’s temperature and luminosity determine its position on the diagram.
2. Stars with the same radius lie along diagonal lines because luminosity depends on both temperature and surface area, as described by the Stefan-Boltzmann law: $$L=\sigma AT^4$$ where
   1. $L$ is the luminosity ($\mathrm{W}$)
   2. $A$ is the surface area ($\mathrm{m}^2$)
   3. $T$ is the absolute temperature of the body ($K$)
   4. $\sigma$ is the Stefan-Boltzmann constant ($\frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}^4}$)