Classification of Stars and Cosmology

Classification of Stars and Cosmology

Info: Board Coverage AQA Paper 2 (Astrophysics option) | Edexcel Paper 1 (P10) | OCR (A) Paper 2 | CIE P4

1. Stellar Classification

Stars are classified by their surface temperature into spectral classes O B A F G K M (hot to cool, remembered by “Oh Be A Fine Girl, Kiss Me”).

The classification is based on the absorption lines in the stellar spectrum: O-type stars show ionised helium lines (hot enough to ionise helium); M-type stars show molecular bands and neutral metals (cool enough for molecules to survive).

The Hydrogen Balmer Series

The Balmer absorption lines (visible) are strongest in A-class stars. In O-type stars the hydrogen is ionised (no bound electron to absorb); in M-type stars the electron is in the $n = 2$ level only rarely (too cold for thermal excitation). A-type stars have the right temperature for significant $n = 2$ population — the sweet spot for Balmer absorption.

2. The Hertzsprung–Russell Diagram

The H–R diagram plots absolute magnitude (or luminosity) against temperature (or spectral class), temperature decreasing left to right.

Key regions:

• Main sequence: stars fusing hydrogen in their cores, running from hot bright O-stars (top left) to cool dim M-dwarfs (bottom right). About 90% of stars.
• Red giants and supergiants: evolved stars that have exhausted core hydrogen; large radius, cool surface, high luminosity (giants) or extreme luminosity (supergiants). Top right.
• White dwarfs: the exposed core of a dead low-mass star; hot but small, so low luminosity despite high temperature. Bottom left.

The Sun is a G-type main-sequence star at absolute magnitude $M = 4.83$.

Stefan–Boltzmann and Stellar Radii

comparing a star to the Sun (subscript $\odot$):

Worked example. Betelgeuse: $T = 3500\ \mathrm{K}$ (about $T_\odot/2$), $L \approx 10^5 L_\odot$. Find its radius.

Betelgeuse is over 1000 solar radii — a supergiant that would engulf the orbit of Jupiter.

3. Supernovae, Neutron Stars and Black Holes

Type II Supernova

A star with mass $> 8 M_\odot$ fuses progressively heavier elements (H → He → C → O → Si → Fe). Iron-56 has the highest binding energy per nucleon, so fusion stops: the core collapses in milliseconds, the outer layers rebound off the rigid core in a shock wave, and the star explodes as a Type II supernova, briefly outshining its entire galaxy.

Neutron Stars

For core masses between roughly 1.4 and 3 $M_\odot$, the collapse halts when neutron degeneracy pressure balances gravity. The result is a neutron star: diameter ~20 km, mass 1.5 $M_\odot$, density comparable to an atomic nucleus ($10^{17}$ kg/m³). Rotation rates up to hundreds of Hz (pulsars).

Black Holes

For core masses $> 3 M_\odot$, not even neutron degeneracy pressure can resist gravity. The core collapses to a black hole: a region where escape velocity exceeds $c$. The boundary is the event horizon at the Schwarzschild radius:

For $M = 10 M_\odot$: $R_s \approx 30$ km.

4. Standard Candles and Distance

Type Ia Supernovae

Type Ia supernovae occur when a white dwarf in a binary system accretes matter from its companion until it reaches the Chandrasekhar limit (~$1.4 M_\odot$) and carbon fusion ignites explosively. Because the mass at detonation is always the same, the peak luminosity is always approximately the same — Type Ia supernovae are standard candles.

The distance is found by comparing the known absolute magnitude ($M \approx -19.3$) to the apparent magnitude $m$ via the distance modulus:

Cepheid Variables

Cepheid variables pulsate with a period that correlates directly with their absolute magnitude (the period–luminosity relationship). Henrietta Leavitt discovered this in 1908; Hubble used it to measure the distance to Andromeda and prove galaxies exist beyond the Milky Way.

5. Hubble’s Law and the Expanding Universe

Hubble’s 1929 observations showed that the redshift of distant galaxies is proportional to their distance:

The redshift $z$ is defined as $z = \frac{\lambda_{\text{obs}} - \lambda_0}{\lambda_0}$, and for $z \ll 1$: $v \approx cz$.

Worked example. A galaxy shows a spectral line at 500.2 nm where the rest wavelength is 500.0 nm. Find the recession velocity and distance.

6. The Big Bang and the CMB

If galaxies are receding, they were closer in the past. Extrapolating backwards leads to the Big Bang model: the universe began from an extremely hot, dense state approximately 13.8 billion years ago.

Evidence:

• Hubble’s law — the universal recession of distant galaxies
• Cosmic Microwave Background (CMB) — predicted by Gamow (1948), discovered by Penzias and Wilson (1965), a uniform 2.7 K black-body radiation filling all space
• Abundance of light elements — Big Bang nucleosynthesis correctly predicts the ~75% H, ~25% He ratio

7. Dark Matter and Dark Energy

Rotation curves of galaxies remain flat at large radii instead of declining as $v \propto r^{-1/2}$ — visible mass alone cannot explain this. The missing mass is called dark matter (does not emit, absorb or reflect light; detected only gravitationally).

Type Ia supernova data from 1998 onward shows that distant supernovae are dimmer than expected: the expansion of the universe is accelerating. The cause is called dark energy ($\Lambda \approx 0.7$ of the total energy density).

The composition of the universe: approximately 5% ordinary matter, 27% dark matter, 68% dark energy.

8. Common Pitfalls

• Magnitude sign convention: brighter stars have more negative magnitudes. The distance modulus $m - M = 5\log_{10}(d/10)$ gives positive $m - M$ for distant (dim-looking) objects.
• Temperature axis on H–R diagram increases to the LEFT (cool right, hot left) — the counterintuitive convention causes diagram-reading errors.
• Type Ia vs Type II: Type Ia = white dwarf + accretion = standard candle. Type II = massive star core collapse = NOT a standard candle (varies too much).
• Redshift formula only valid for $z \ll 1$: at large $z$, relativistic cosmology replaces the simple $v = cz$ relation.
• The CMB temperature (2.7 K) is the current temperature: it has cooled from ~3000 K at recombination due to expansion.

9. Summary

See Also

• Nuclear Physics — fusion and binding energy
• Particle Physics — the standard model
• Gravitational Fields — gravitational potential and escape velocity