§ 25.1 Standard candles
Key ideas
- Luminosity L is the total power a star radiates; the radiant flux intensity F is the power per unit area reaching us.
- The flux obeys an inverse-square law as the light spreads over a sphere of radius d.
- A standard candle has a known luminosity, so measuring its flux gives its distance.
Equations
F = L / (4π d²)flux from luminosity and distanceW m⁻²
Fig. 1 · The inverse-square law: the same power L spreads over ever larger spheres, so the flux through a fixed area falls as 1/d² with distance.
Watch out: luminosity L is intrinsic to the star and the same for every observer; the flux F depends on distance. A dim-looking star may simply be far away, not faint.
§ 25.2 Stellar radii
Key ideas
- Wien's law: the peak wavelength of a star's radiation is inversely proportional to its temperature, so hotter stars look bluer.
- The Stefan-Boltzmann law links luminosity to surface area and temperature; with L and T known, the radius follows.
Equations
λ_max T = 2.9 × 10⁻³ m KWien's displacement lawm K
L = 4π σ r² T⁴σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴W
Fig. 2 · Hotter stars (solid) radiate more at every wavelength and peak at a shorter λ_max; Wien's law reads the temperature off the peak, Stefan-Boltzmann gives the radius.
Watch out: luminosity depends on T⁴, so a small temperature difference is a huge luminosity difference. Doubling a star's temperature multiplies its power output sixteenfold.
§ 25.3 Hubble's law and the Big Bang
Key ideas
- Light from receding galaxies is redshifted; for speeds well below c the fractional shift gives the recession speed.
- Hubble's law: the recession speed is proportional to distance, so distant galaxies recede faster.
- This points to an expanding Universe from a hot, dense Big Bang; the age is estimated as about 1/H₀.
Equations
Δλ/λ ≈ v / credshift for a slowly receding galaxyno unit
v = H₀ dHubble's law; age ≈ 1/H₀km s⁻¹
Fig. 3 · Hubble's law: a straight line through the origin, recession speed against distance, whose gradient is the Hubble constant H₀; its reciprocal estimates the age of the Universe.
Watch out: to get an age from 1/H₀ you must put H₀ in SI units (s⁻¹). Convert from km s⁻¹ Mpc⁻¹ first (1 Mpc ≈ 3.09 × 10¹⁹ km), or the answer is wildly wrong.